This note provides the some sum formulas for generalized Fibonacci numbers. The results are proved using clever rearrangements, rather than using induction.
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.
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 present some identities for the sums of squares of Fibonacci and Lucas numbers with consecutive primes, using maximal prime gap G(x)~log2x, as indices.
A survey of zoological literature affirmed the wide occurrence of Fibonacci numbers in the organization of acellular and prokaryotic life forms as well as in some eukaryotic protistans and in the embryonic development...A survey of zoological literature affirmed the wide occurrence of Fibonacci numbers in the organization of acellular and prokaryotic life forms as well as in some eukaryotic protistans and in the embryonic development and adult forms of many living and fossil remains of metazoan animals. A detailed comparative analysis of the axial skeleton of a fossil fish and humans revealed a new rule of the “nested triad” of bones organized along the proximal to distal axis of limb appendages. This growth pattern and its ubiquity among living vertebrates appear to underlie a profound rule of pattern formation that is dictated in part by the genetics and epigenetic mechanisms of stem cell clonal development.展开更多
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
Fibonacci number spiral patterns can be found in nature,particularly in plants,such as the sunflowers and phyllotaxis.Here,we demonstrated this pattern can be reproduced spontaneously within self-assembling peptide na...Fibonacci number spiral patterns can be found in nature,particularly in plants,such as the sunflowers and phyllotaxis.Here,we demonstrated this pattern can be reproduced spontaneously within self-assembling peptide nanofibril films.By high-temperature water vapor annealing of an amorphous film containing both peptide and cationic diamines,well-defined amyloid-like nanofibrils can be assembled spontaneously,during which the nanofibrils will hierarchically stack with each other following the Fibonacci number patterns.The formation of the patterns is a selftemplated process,which involves stepwise chiral amplification from the molecular scale to the nano-and micro-scales.Moreover,by controlling the diameter,length,and handedness of the nanofibrils,various complex hierarchical structures could be formed,including vertically aligned nanoarray,mesoscale helical bundles,Fibonacci number spirals,and then helical toroids.The results provide new insights into the chiral self-assembly of simple biological molecules,which can advance their applications in optics and templated synthesis.展开更多
Previously, synthetic hexagonal bismuth sulfide iodide (polar space group P63, a = 15.629(3) ?, c = 4.018(1) ?, Z = 2) has been described by the rather unsatisfactory fractional formula Bi19/3IS9 [1] 08D0C9EA79F9BACE1...Previously, synthetic hexagonal bismuth sulfide iodide (polar space group P63, a = 15.629(3) ?, c = 4.018(1) ?, Z = 2) has been described by the rather unsatisfactory fractional formula Bi19/3IS9 [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310037003600350038003400370039000000 -[3] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310037003600350038003400370036000000 . A redetermination of the structure using old but reliable photographic intensity data indicated the presence of additional split positions and reduced atomic occupancies. From the observed pattern of this “averaged” structure a consistent model of a superstructure with lattice parameters of a' = √13·a = 56.35(1) ?, c' = c, and a formula Bi5-x(Bi2S3)39I12S emerged, with 2 formula units in a cell of likewise P63 space group. Structural modulation may be provoked by the space the lone electron pair of Bi requires. When Bi on the 0, 0, z position of the “averaged” cell is transferred to two general six-fold sites and one unoccupied twofold one of the super-cell, more structural stability is guaranteed due to compensation of its basal plane dipole momentum. Owing to the limited intensity data available, more details of the superstructure are not accessible yet. Some physical properties and solar cell application are discussed together with suggestions of ambient temperature synthesis routes of c-axis oriented nano-rod sheets.展开更多
The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the hist...The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the historical backgrounds of these concepts, we propose an algorithm that will generate the elements of these matrices. In fact, we will show that the indicated algorithm can be used to construct the elements of any power series matrix generated by any polynomial (see Definition 1), and hence, it is a generalization of the specific algorithms that give us the Pascal and the Fibonacci matrices.展开更多
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 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.
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.
文摘In this paper we present some identities for the sums of squares of Fibonacci and Lucas numbers with consecutive primes, using maximal prime gap G(x)~log2x, as indices.
文摘A survey of zoological literature affirmed the wide occurrence of Fibonacci numbers in the organization of acellular and prokaryotic life forms as well as in some eukaryotic protistans and in the embryonic development and adult forms of many living and fossil remains of metazoan animals. A detailed comparative analysis of the axial skeleton of a fossil fish and humans revealed a new rule of the “nested triad” of bones organized along the proximal to distal axis of limb appendages. This growth pattern and its ubiquity among living vertebrates appear to underlie a profound rule of pattern formation that is dictated in part by the genetics and epigenetic mechanisms of stem cell clonal development.
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(21621004 and 22078239)Tianjin Development Program for Innovation and Entrepreneurship(2018)the State Key Laboratory of Chemical Engineering(SKL-Ch E-20Z04 and SKL-Ch E-21T03)。
文摘Fibonacci number spiral patterns can be found in nature,particularly in plants,such as the sunflowers and phyllotaxis.Here,we demonstrated this pattern can be reproduced spontaneously within self-assembling peptide nanofibril films.By high-temperature water vapor annealing of an amorphous film containing both peptide and cationic diamines,well-defined amyloid-like nanofibrils can be assembled spontaneously,during which the nanofibrils will hierarchically stack with each other following the Fibonacci number patterns.The formation of the patterns is a selftemplated process,which involves stepwise chiral amplification from the molecular scale to the nano-and micro-scales.Moreover,by controlling the diameter,length,and handedness of the nanofibrils,various complex hierarchical structures could be formed,including vertically aligned nanoarray,mesoscale helical bundles,Fibonacci number spirals,and then helical toroids.The results provide new insights into the chiral self-assembly of simple biological molecules,which can advance their applications in optics and templated synthesis.
文摘Previously, synthetic hexagonal bismuth sulfide iodide (polar space group P63, a = 15.629(3) ?, c = 4.018(1) ?, Z = 2) has been described by the rather unsatisfactory fractional formula Bi19/3IS9 [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310037003600350038003400370039000000 -[3] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310037003600350038003400370036000000 . A redetermination of the structure using old but reliable photographic intensity data indicated the presence of additional split positions and reduced atomic occupancies. From the observed pattern of this “averaged” structure a consistent model of a superstructure with lattice parameters of a' = √13·a = 56.35(1) ?, c' = c, and a formula Bi5-x(Bi2S3)39I12S emerged, with 2 formula units in a cell of likewise P63 space group. Structural modulation may be provoked by the space the lone electron pair of Bi requires. When Bi on the 0, 0, z position of the “averaged” cell is transferred to two general six-fold sites and one unoccupied twofold one of the super-cell, more structural stability is guaranteed due to compensation of its basal plane dipole momentum. Owing to the limited intensity data available, more details of the superstructure are not accessible yet. Some physical properties and solar cell application are discussed together with suggestions of ambient temperature synthesis routes of c-axis oriented nano-rod sheets.
文摘The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the historical backgrounds of these concepts, we propose an algorithm that will generate the elements of these matrices. In fact, we will show that the indicated algorithm can be used to construct the elements of any power series matrix generated by any polynomial (see Definition 1), and hence, it is a generalization of the specific algorithms that give us the Pascal and the Fibonacci matrices.
