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Quasi-Fibonacci numbers of the seventh order. (English) Zbl 1178.11028

Summary: In this paper we introduce and investigate the so-called quasi-Fibonacci numbers of the seventh order. We discover many surprising relations and identities, and study some applications to polynomials.

MSC:

11B83 Special sequences and polynomials
11A07 Congruences; primitive roots; residue systems
11B39 Fibonacci and Lucas numbers and polynomials and generalizations

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Online Encyclopedia of Integer Sequences:

Random walks (binomial transform of A006054).
a(n) = 2*a(n-1) + a(n-2) - a(n-3), with a(0) = a(1) = 0, a(2) = 1.
a(n) = 2*a(n-1) + a(n-2) - a(n-3) for n >= 3, starting with a(0) = 1, a(1) = 3, and a(2) = 6.
Expansion of (2 + 2*x - 3*x^2) / (1 - 2*x - x^2 + x^3).
Expansion of (1-x)*(1+x)/(1-2*x-x^2+x^3).
Expansion of (1-2*x)*(1-x)/(1-5*x+6*x^2-x^3).
Expansion of (1-x)/(1-2*x-x^2+x^3).
Number of three-choice paths along a corridor of height 5, starting from the lower side.
An accelerator sequence for Catalan’s constant.
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 4.
Expansion of (1-x^2)/(1-x-9*x^2+x^3).
Expansion of (1 - 3*x + 2*x^2)/(1 - 4*x + 3*x^2 + x^3).
Expansion of (1+x-2*x^2)/(1-21*x^2-7*x^3).
Let i be in {1,2,3} and let r >= 0 be an integer. Let p = {p_1, p_2, p_3} = {-2,0,1}, n=2*r+p_i, and define a(-2)=1. Then, a(n)=a(2*r+p_i) gives the quantity of H_(7,1,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt(2*cos(Pi/7)).
Let i be in {1,2,3} and let r >= 0 be an integer. Let p = {p_1, p_2, p_3} = {-2,0,1}, n=2*r+p_i, and define a(-2)=0. Then, a(n)=a(2*r+p_i) gives the quantity of H_(7,2,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt(2*cos(Pi/7)).
Let i be in {1,2,3} and let r >= 0 be an integer. Let p = {p_1, p_2, p_3} = {-2,0,1}, n = 2*r + p_i and define a(-2)=0. Then, a(n) = a(2*r + p_i) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x = sqrt(2*cos(Pi/7)).
Let i be in {1,2,3}, let r >= 0 be an integer and n=2*r+i-1. Then a(n)=a(2*r+i-1) gives the quantity of H_(7,1,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).
Let i be in {1,2,3}, let r >= 0 be an integer and n=2*r+i-1. Then a(n)=a(2*r+i-1) gives the quantity of H_(7,2,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).
Let i be in {1,2,3}, let r >= 0 be an integer and n=2*r+i-1. Then a(n)=a(2*r+i-1) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).
a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3), with a(0)=0, a(1)=0 and a(2)=1.
a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=a(1)=0, a(2)=9.
a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=0, a(1)=3, and a(2)=6.
a(n) = n^3 + 2*n^2 + 5*n + 11.
Expansion of 2*x*(1 + x)/(1 - x - 9*x^2 + x^3).
Expansion of 4*x^2/(1 - x - 9*x^2 + x^3).
Sum of n-th powers of the roots of x^3 + 9*x^2 - x - 1.
Sum of n-th powers of the roots of x^3 + x^2 - 9*x - 1.
a(n) = (-cos(Pi/7)/cos(2*Pi/7))^n + (-cos(2*Pi/7)/cos(3*Pi/7))^n + (cos(3*Pi/7)/cos(Pi/7))^n.
a(n) = (7*(csc(2*Pi/7))^2)^n + (7*(csc(4*Pi/7))^2)^n + (7*(csc(8*Pi/7))^2)^n.
a(n) is the number of different linear hydrocarbon molecules with n carbon atoms.