By studying the properties of Chebyshev polynomials, some specific and mean-ingful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbersand Lucas numbers are obtained.
This note provides the some sum formulas for generalized Fibonacci numbers. The results are proved using clever rearrangements, rather than using induction.
By applying the method of on summation by parts,the purpose of this paper is to give several reciprocal summations related to squares of products of the Fibonacci numbers.
In this paper,we consider infinite sums of the reciprocals of the Fibonacci numbers.Then applying the floor function to the reciprocals of this sums,we obtain a new identity involving the Fibonacci numbers.
Let us define to be a r-Toeplitz matrix. The entries in the first row of are or;where F<sub>n</sub> and L<sub>n</sub> denote the usual Fibonacci and Lucas numbers, respe...Let us define to be a r-Toeplitz matrix. The entries in the first row of are or;where F<sub>n</sub> and L<sub>n</sub> denote the usual Fibonacci and Lucas numbers, respectively. We obtained some bounds for the spectral norm of these matrices.展开更多
In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important...In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important to mention that results of this nature were given by Santos and Ivkovic in two papers published on the Fibonacci Quarterly, Polynomial generalizations of the Pell sequence and the Fibonacci sequence [1] and Fibonacci Numbers and Partitions [2] , and one, by Santos, on Discrete Mathematics, On the Combinatorics of Polynomial generalizations of Rogers-Ramanujan Type Identities [3]. By these results one can see that from the q-series identities important combinatorial information can be obtained by a careful study of the two variable function introduced by Andrews in Combinatorics and Ramanujan's lost notebook [4].展开更多
Let us define A=Hr=(aij)?to be n×n?r-Hankel matrix. The entries of matrix A are Fn=Fi+j-2?or Ln=Fi+j-2?where Fn?and Ln?denote the usual Fibonacci and Lucas numbers, respectively. Then, we obtained upper and l...Let us define A=Hr=(aij)?to be n×n?r-Hankel matrix. The entries of matrix A are Fn=Fi+j-2?or Ln=Fi+j-2?where Fn?and Ln?denote the usual Fibonacci and Lucas numbers, respectively. Then, we obtained upper and lower bounds for the spectral norm of matrix A. We compared our bounds with exact value of matrix A’s spectral norm. These kinds of matrices have connections with signal and image processing, time series analysis and many other problems.展开更多
The Fibonacci numbers are the numbers defined by the linear recurrence equation, in which each subsequent number is the sum of the previous two. In this paper, we propose Fibonacci networks using Fibonacci numbers. Th...The Fibonacci numbers are the numbers defined by the linear recurrence equation, in which each subsequent number is the sum of the previous two. In this paper, we propose Fibonacci networks using Fibonacci numbers. The analyticai expressions involving degree distribution, average path lengh and mean first passage time are obtained. This kind of networks exhibits the smail-world characteristic and follows the exponential distribution. Our proposed models would provide the vaiuable insights into the deterministicaily delayed growing networks.展开更多
The Advanced Encryption Standard(AES)is the most widely used symmetric cipher today.AES has an important place in cryptology.Finite field,also known as Galois Fields,are cornerstones for understanding any cryptography...The Advanced Encryption Standard(AES)is the most widely used symmetric cipher today.AES has an important place in cryptology.Finite field,also known as Galois Fields,are cornerstones for understanding any cryptography.This encryption method on AES is a method that uses polynomials on Galois fields.In this paper,we generalize the AES-like cryptology on 2×2 matrices.We redefine the elements of k-order Fibonacci polynomials sequences using a certain irreducible polynomial in our cryptology algorithm.So,this cryptology algorithm is called AES-like cryptology on the k-order Fibonacci polynomial matrix.展开更多
The empirical relation of between the transition temperature of optimum doped superconductors T<sub>co</sub> and the mean cationic charge , a physical paradox, can be recast to strongly support fractal the...The empirical relation of between the transition temperature of optimum doped superconductors T<sub>co</sub> and the mean cationic charge , a physical paradox, can be recast to strongly support fractal theories of high-T<sub>c</sub> superconductors, thereby applying the finding that the optimum hole concentration of σ<sub>o</sub> = 0.229 can be linked with the universal fractal constant δ<sub>1</sub> = 8.72109… of the renormalized quadratic Hénon map. The transition temperature obviously increases steeply with a domain structure of ever narrower size, characterized by Fibonacci numbers. However, also conventional BCS superconductors can be scaled with δ<sub>1</sub>, exemplified through the energy gap relation k<sub>B</sub>T<sub>c</sub> ≈ 5Δ<sub>0</sub>/δ<sub>1</sub>, suggesting a revision of the entire theory of superconductivity. A low mean cationic charge allows the development of a frustrated nano-sized fractal structure of possibly ferroelastic nature delivering nano-channels for very fast charge transport, in common for both high-T<sub>c</sub> superconductor and organic-inorganic halide perovskite solar materials. With this backing superconductivity above room temperature can be conceived for synthetic sandwich structures of less than 2+. For instance, composites of tenorite and cuprite respectively tenorite and CuI (CuBr, CuCl) onto AuCu alloys are proposed. This specification is suggested by previously described filamentary superconductivity of “bulk” CuO1﹣x samples. In addition, cesium substitution in the Tl-1223 compound is an option.展开更多
Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is ...Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is and how to generate them entirely. We also show which kinds of integers have the maximum number of optimal BSD-Rs.展开更多
This article presents some new results on the class SLMα of functions that are analytic in the open unit disc U = {z ： ｜z｜〈 1} satisfying the conditions that f（0） =0, f＇（0）= 1, and α（1＋zf＇＇（z）/f＇（...This article presents some new results on the class SLMα of functions that are analytic in the open unit disc U = {z ： ｜z｜〈 1} satisfying the conditions that f（0） =0, f＇（0）= 1, and α（1＋zf＇＇（z）/f＇（z）＋（1-α）zf＇（z）/f（z）∈p（U）for all z ∈ U, where αis a real number and p（z）=1＋r^2z^2/1-Tz-T^2z^2（z∈ U）. The number T = （1 -√5）/2 is such that T^2 = 1 ＋ T. The class SFLMa introduced by J. Dziok, R.K. Raina, and J. Sokot [3, Appl. Math. Comput. 218 （2011）, 996-1002] is closely related to the classes of starlike and convex functions. The article deals with several ideas and techniques used in geometric function theory and differential subordinations theory.展开更多
Given a Riordan array,its vertical half and horizontal half are studied separately before.In the present paper,we introduce the(m,r,s)-halves of a Riordan array.This allows us to discuss the vertical half and horizont...Given a Riordan array,its vertical half and horizontal half are studied separately before.In the present paper,we introduce the(m,r,s)-halves of a Riordan array.This allows us to discuss the vertical half and horizontal half in a uniform context.As applications,we find several new identities involving Fibonacci,Pell and Jacobsthal sequences by applying the(m,r,s)-halves of Pascal and Delannoy matrices.展开更多
Synchronicity involves the experience of personal meaning entangled with ambiguous coincidences in time. Ambiguity results from incomplete information about the chances of various events occurring. The problem that th...Synchronicity involves the experience of personal meaning entangled with ambiguous coincidences in time. Ambiguity results from incomplete information about the chances of various events occurring. The problem that this study addresses is the lack of empirical research on synchronicity. This study sought to address this problem by exploring the astrological hypothesis that planetary transits predict synchronicity events. Synchronicities were compared with the probability distributions of planetary transits. In comparison with the base rate prediction, planetary transits were not a significant predictor of synchronicity events. The findings of this study provide new insight into the complex, multifaceted, and ambiguous phenomenon of synchronicity. The concept of ambiguity tolerance plays a significant role in synchronicity research since ambiguity cannot be completely eliminated.展开更多
The problem of determining the number of steps needed to find the greatest common divisor of two positive integers by Euclidean algorithm has been investigated in elementary number theory for decades. Different upper ...The problem of determining the number of steps needed to find the greatest common divisor of two positive integers by Euclidean algorithm has been investigated in elementary number theory for decades. Different upper bounds have been found for this problem. Here, we provide a sharp upper bound for a function which has a direct relation to the numbers whom the greatest common divisor we are trying to calculate. We mainly use some features of Fibonacci numbers as our tools.展开更多
We study the eight infinite sequences of triples of natural numbers A=(F2n+1,4F2n+3,F2n+7), B=(F2n+1,4F2n+5,F2n+7), C=(F2n+1,5F2n+1,F2n+3), D=(F2n+3,4F2n+1,F2n+3) and A=(L2n+1,4L2n+3,L2n+7), B=(L2n+1,4L2n+5,L2n+7), C=...We study the eight infinite sequences of triples of natural numbers A=(F2n+1,4F2n+3,F2n+7), B=(F2n+1,4F2n+5,F2n+7), C=(F2n+1,5F2n+1,F2n+3), D=(F2n+3,4F2n+1,F2n+3) and A=(L2n+1,4L2n+3,L2n+7), B=(L2n+1,4L2n+5,L2n+7), C=(L2n+1,5L2n+1,L2n+3), D=(L2n+3,4L2n+1,L2n+3. The sequences A,B,C and D are built from the Fibonacci numbers Fn while the sequences A, B, C and D from the Lucas numbers Ln. Each triple in the sequences A,B,C and D has the property D(-4) (i. e., adding -4 to the product of any two different components of them is a square). Similarly, each triple in the sequences A, B, C and D has the property D(20). We show some interesting properties of these sequences that give various methods how to get squares from them.展开更多
In this paper, we first present several identities related to the inverse-conjugate compositions having parts of size ≤ 3, the compositions into parts equal to 1 or 2, the compositions into odd parts and the composit...In this paper, we first present several identities related to the inverse-conjugate compositions having parts of size ≤ 3, the compositions into parts equal to 1 or 2, the compositions into odd parts and the compositions into parts greater than 1. In addition, we provide a bijective proof of a relation for inverse-conjugate compositions having parts of size ≤ k.展开更多
A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic...A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic one, but is connected with the inverse of Sommerfeld’s fine-structure constant and this way again connected with the electron. From number-theoretical realities, including the reciprocity relation of the golden ratio as effective pre-calculator of nature’s creativeness, a proposed closeness to the icosahedron may point towards the structure of the electron, thought off as a single-strand compacted helically self-confined charged elemantary particle of less spherical but assumed blunted icosahedral shape generated from a high energy double-helix photon. We constructed a chiral Moebius “ball” from a 13 times 180˚twisted double helix strand, where the turning points of 12 generated slings were arranged towards the vertices of a regular icosahedron, belonging to the non-centrosymmetric rotation group I532. Mathematically put, we convert the helical motion of an energy quantum into a stationary motion on a Moebius stripe structure. The radius of the ball is about the Compton radius. This chiral closed circuit Moebius ball motion profile can be tentatively thought off as the dominant quantum vortex structure of the electron, and the model may be named CEWMB (Charged Electromagnetic Wave Moebius Ball). Also the gyromagnetic factor of the electron (g<sub>e</sub> = 2.002319) can be traced back to this special structure. However, nature’s energy infinity principle would suggest a superposition with additional less dominant (secondary) structures, governed also by the golden mean. A suggestion about the possible structure of delocalized hole carriers in the superconducting state is given.展开更多
In this paper, we study the palindromic compositions of even integers when no 2＇s are allowed in a composition and its conjugate. We show that the number of these palindromes is equal to 2Fn-1, where, Fn is the n-th ...In this paper, we study the palindromic compositions of even integers when no 2＇s are allowed in a composition and its conjugate. We show that the number of these palindromes is equal to 2Fn-1, where, Fn is the n-th Fibonacci number. Consequently, we obtain several identities between the number of these palindromes, the number of compositions into parts equal to 1＇s or 2＇s, the number of compositions into odd parts and the number of compositions into parts greater than 1.展开更多
Chinese calligraphy is a thousand-year-old writing art. The question of how Chinese calligraphy artworks convey emotion has cast its spell over people for millennia. Calligraphers' joys and sorrows were expressed ...Chinese calligraphy is a thousand-year-old writing art. The question of how Chinese calligraphy artworks convey emotion has cast its spell over people for millennia. Calligraphers' joys and sorrows were expressed in the complexity of the character strokes, style variations and general layouts. Determining how Chinese calligraphy aesthetic patterns emerged from the general layout of artworks is a challenging objective for researchers. Here we investigate the statistical fluctuation structure of Chinese calligraphy characters sizes using characters obtained from the calligraphy artwork "Preface to the Poems Collected from the Orchid Pavilion" which was praised as the best running script under heaven. We found that the character size distribution is a stretched exponential distribution. Moreover, the variations in the local correlation features in character size fluctuations can accurately reflect expressions of the calligrapher's complex feelings. The fractal dimensions of character size fluctuations are close to the Fibonacci sequence. The Fibonacci number is first discovered in the Chinese calligraphy artworks, which inspires the aesthetics of Chinese calligraphy artworks and maybe also provides an approach to creating Chinese calligraphy artworks in multiple genres.展开更多
基金Supported by the Natural Science Foundation of Shaanxi Province(2002A11)Supported by the Shangluo Teacher's College Research Foundation(SKY2106)
文摘By studying the properties of Chebyshev polynomials, some specific and mean-ingful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbersand Lucas numbers are obtained.
文摘This note provides the some sum formulas for generalized Fibonacci numbers. The results are proved using clever rearrangements, rather than using induction.
基金Supported by the Natural Science Foundation of Henan Province(0511010300)Supported by the Natural Science Foundation of Education Department of Henan Province(2008B110011)
文摘By applying the method of on summation by parts,the purpose of this paper is to give several reciprocal summations related to squares of products of the Fibonacci numbers.
基金Supported by the National Natural Science Foundation of China (Grant No.11071194)
文摘In this paper,we consider infinite sums of the reciprocals of the Fibonacci numbers.Then applying the floor function to the reciprocals of this sums,we obtain a new identity involving the Fibonacci numbers.
文摘Let us define to be a r-Toeplitz matrix. The entries in the first row of are or;where F<sub>n</sub> and L<sub>n</sub> denote the usual Fibonacci and Lucas numbers, respectively. We obtained some bounds for the spectral norm of these matrices.
基金Partially supported by FAPESP(Fundacao de Amparo a Pesquisa do Estado de Sao Paulo).
文摘In this paper we present combinatorial interpretations and polynomials generalizations for sequences including the Fibonacci numbers, the Pell numbers and the Jacobsthal numbers in terms of partitions. It is important to mention that results of this nature were given by Santos and Ivkovic in two papers published on the Fibonacci Quarterly, Polynomial generalizations of the Pell sequence and the Fibonacci sequence [1] and Fibonacci Numbers and Partitions [2] , and one, by Santos, on Discrete Mathematics, On the Combinatorics of Polynomial generalizations of Rogers-Ramanujan Type Identities [3]. By these results one can see that from the q-series identities important combinatorial information can be obtained by a careful study of the two variable function introduced by Andrews in Combinatorics and Ramanujan's lost notebook [4].
文摘Let us define A=Hr=(aij)?to be n×n?r-Hankel matrix. The entries of matrix A are Fn=Fi+j-2?or Ln=Fi+j-2?where Fn?and Ln?denote the usual Fibonacci and Lucas numbers, respectively. Then, we obtained upper and lower bounds for the spectral norm of matrix A. We compared our bounds with exact value of matrix A’s spectral norm. These kinds of matrices have connections with signal and image processing, time series analysis and many other problems.
基金Supported by the National Natural Science Foundation of China under Grant No.61203155Zhejiang Provincial Natural Science Foundation under Grant No.LQ12F03003
文摘The Fibonacci numbers are the numbers defined by the linear recurrence equation, in which each subsequent number is the sum of the previous two. In this paper, we propose Fibonacci networks using Fibonacci numbers. The analyticai expressions involving degree distribution, average path lengh and mean first passage time are obtained. This kind of networks exhibits the smail-world characteristic and follows the exponential distribution. Our proposed models would provide the vaiuable insights into the deterministicaily delayed growing networks.
基金This work is supported by the Scientific Research Project(BAP)2020FEBE009,Pamukkale University,Denizli,Turkey.
文摘The Advanced Encryption Standard(AES)is the most widely used symmetric cipher today.AES has an important place in cryptology.Finite field,also known as Galois Fields,are cornerstones for understanding any cryptography.This encryption method on AES is a method that uses polynomials on Galois fields.In this paper,we generalize the AES-like cryptology on 2×2 matrices.We redefine the elements of k-order Fibonacci polynomials sequences using a certain irreducible polynomial in our cryptology algorithm.So,this cryptology algorithm is called AES-like cryptology on the k-order Fibonacci polynomial matrix.
文摘The empirical relation of between the transition temperature of optimum doped superconductors T<sub>co</sub> and the mean cationic charge , a physical paradox, can be recast to strongly support fractal theories of high-T<sub>c</sub> superconductors, thereby applying the finding that the optimum hole concentration of σ<sub>o</sub> = 0.229 can be linked with the universal fractal constant δ<sub>1</sub> = 8.72109… of the renormalized quadratic Hénon map. The transition temperature obviously increases steeply with a domain structure of ever narrower size, characterized by Fibonacci numbers. However, also conventional BCS superconductors can be scaled with δ<sub>1</sub>, exemplified through the energy gap relation k<sub>B</sub>T<sub>c</sub> ≈ 5Δ<sub>0</sub>/δ<sub>1</sub>, suggesting a revision of the entire theory of superconductivity. A low mean cationic charge allows the development of a frustrated nano-sized fractal structure of possibly ferroelastic nature delivering nano-channels for very fast charge transport, in common for both high-T<sub>c</sub> superconductor and organic-inorganic halide perovskite solar materials. With this backing superconductivity above room temperature can be conceived for synthetic sandwich structures of less than 2+. For instance, composites of tenorite and cuprite respectively tenorite and CuI (CuBr, CuCl) onto AuCu alloys are proposed. This specification is suggested by previously described filamentary superconductivity of “bulk” CuO1﹣x samples. In addition, cesium substitution in the Tl-1223 compound is an option.
基金Supported by Chinese National Basic Research Program(2007CB807902)
文摘Binary signed digit representation （BSD-R） of an integer is widely used in computer arithmetic, cryptography and digital signal processing. This paper studies what the exact number of optimal BSD-R of an integer is and how to generate them entirely. We also show which kinds of integers have the maximum number of optimal BSD-Rs.
文摘This article presents some new results on the class SLMα of functions that are analytic in the open unit disc U = {z ： ｜z｜〈 1} satisfying the conditions that f（0） =0, f＇（0）= 1, and α（1＋zf＇＇（z）/f＇（z）＋（1-α）zf＇（z）/f（z）∈p（U）for all z ∈ U, where αis a real number and p（z）=1＋r^2z^2/1-Tz-T^2z^2（z∈ U）. The number T = （1 -√5）/2 is such that T^2 = 1 ＋ T. The class SFLMa introduced by J. Dziok, R.K. Raina, and J. Sokot [3, Appl. Math. Comput. 218 （2011）, 996-1002] is closely related to the classes of starlike and convex functions. The article deals with several ideas and techniques used in geometric function theory and differential subordinations theory.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1210128011861045)the Science Foundation for Youths of Gansu Province(Grant No.20JR10RA187)。
文摘Given a Riordan array,its vertical half and horizontal half are studied separately before.In the present paper,we introduce the(m,r,s)-halves of a Riordan array.This allows us to discuss the vertical half and horizontal half in a uniform context.As applications,we find several new identities involving Fibonacci,Pell and Jacobsthal sequences by applying the(m,r,s)-halves of Pascal and Delannoy matrices.
文摘Synchronicity involves the experience of personal meaning entangled with ambiguous coincidences in time. Ambiguity results from incomplete information about the chances of various events occurring. The problem that this study addresses is the lack of empirical research on synchronicity. This study sought to address this problem by exploring the astrological hypothesis that planetary transits predict synchronicity events. Synchronicities were compared with the probability distributions of planetary transits. In comparison with the base rate prediction, planetary transits were not a significant predictor of synchronicity events. The findings of this study provide new insight into the complex, multifaceted, and ambiguous phenomenon of synchronicity. The concept of ambiguity tolerance plays a significant role in synchronicity research since ambiguity cannot be completely eliminated.
文摘The problem of determining the number of steps needed to find the greatest common divisor of two positive integers by Euclidean algorithm has been investigated in elementary number theory for decades. Different upper bounds have been found for this problem. Here, we provide a sharp upper bound for a function which has a direct relation to the numbers whom the greatest common divisor we are trying to calculate. We mainly use some features of Fibonacci numbers as our tools.
文摘We study the eight infinite sequences of triples of natural numbers A=(F2n+1,4F2n+3,F2n+7), B=(F2n+1,4F2n+5,F2n+7), C=(F2n+1,5F2n+1,F2n+3), D=(F2n+3,4F2n+1,F2n+3) and A=(L2n+1,4L2n+3,L2n+7), B=(L2n+1,4L2n+5,L2n+7), C=(L2n+1,5L2n+1,L2n+3), D=(L2n+3,4L2n+1,L2n+3. The sequences A,B,C and D are built from the Fibonacci numbers Fn while the sequences A, B, C and D from the Lucas numbers Ln. Each triple in the sequences A,B,C and D has the property D(-4) (i. e., adding -4 to the product of any two different components of them is a square). Similarly, each triple in the sequences A, B, C and D has the property D(20). We show some interesting properties of these sequences that give various methods how to get squares from them.
基金Supported by the National Natural Science Foundation of China(Grant No.11461020)
文摘In this paper, we first present several identities related to the inverse-conjugate compositions having parts of size ≤ 3, the compositions into parts equal to 1 or 2, the compositions into odd parts and the compositions into parts greater than 1. In addition, we provide a bijective proof of a relation for inverse-conjugate compositions having parts of size ≤ k.
文摘A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic one, but is connected with the inverse of Sommerfeld’s fine-structure constant and this way again connected with the electron. From number-theoretical realities, including the reciprocity relation of the golden ratio as effective pre-calculator of nature’s creativeness, a proposed closeness to the icosahedron may point towards the structure of the electron, thought off as a single-strand compacted helically self-confined charged elemantary particle of less spherical but assumed blunted icosahedral shape generated from a high energy double-helix photon. We constructed a chiral Moebius “ball” from a 13 times 180˚twisted double helix strand, where the turning points of 12 generated slings were arranged towards the vertices of a regular icosahedron, belonging to the non-centrosymmetric rotation group I532. Mathematically put, we convert the helical motion of an energy quantum into a stationary motion on a Moebius stripe structure. The radius of the ball is about the Compton radius. This chiral closed circuit Moebius ball motion profile can be tentatively thought off as the dominant quantum vortex structure of the electron, and the model may be named CEWMB (Charged Electromagnetic Wave Moebius Ball). Also the gyromagnetic factor of the electron (g<sub>e</sub> = 2.002319) can be traced back to this special structure. However, nature’s energy infinity principle would suggest a superposition with additional less dominant (secondary) structures, governed also by the golden mean. A suggestion about the possible structure of delocalized hole carriers in the superconducting state is given.
基金Supported by the National Natural Science Foundation of China(Grant No.11461020)
文摘In this paper, we study the palindromic compositions of even integers when no 2＇s are allowed in a composition and its conjugate. We show that the number of these palindromes is equal to 2Fn-1, where, Fn is the n-th Fibonacci number. Consequently, we obtain several identities between the number of these palindromes, the number of compositions into parts equal to 1＇s or 2＇s, the number of compositions into odd parts and the number of compositions into parts greater than 1.
基金funded by the National Natural Science Foundation of China (No. 41465010, 41977245)。
文摘Chinese calligraphy is a thousand-year-old writing art. The question of how Chinese calligraphy artworks convey emotion has cast its spell over people for millennia. Calligraphers' joys and sorrows were expressed in the complexity of the character strokes, style variations and general layouts. Determining how Chinese calligraphy aesthetic patterns emerged from the general layout of artworks is a challenging objective for researchers. Here we investigate the statistical fluctuation structure of Chinese calligraphy characters sizes using characters obtained from the calligraphy artwork "Preface to the Poems Collected from the Orchid Pavilion" which was praised as the best running script under heaven. We found that the character size distribution is a stretched exponential distribution. Moreover, the variations in the local correlation features in character size fluctuations can accurately reflect expressions of the calligrapher's complex feelings. The fractal dimensions of character size fluctuations are close to the Fibonacci sequence. The Fibonacci number is first discovered in the Chinese calligraphy artworks, which inspires the aesthetics of Chinese calligraphy artworks and maybe also provides an approach to creating Chinese calligraphy artworks in multiple genres.
