A367456 Expansion of (1 - x)/(1 - x - 7*x^2).

1, 0, 7, 7, 56, 105, 497, 1232, 4711, 13335, 46312, 139657, 463841, 1441440, 4688327, 14778407, 47596696, 151045545, 484222417, 1541541232, 4931098151, 15721886775, 50239573832, 160292781257, 511969798081, 1634019266880, 5217807853447, 16655942721607, 53180597695736, 169772196746985

Offset: 0

Author

Wolfdieter Lang, Jan 16 2024

Keywords

• nonn
• easy

Comments

a(n) appears in the formula for powers of the fundamental algebraic number c = (1 + sqrt(29))/2 = A223140 of the quadratic number field Q(sqrt(29)): c^n = a(n) + A015442(n), for n >= 0. The formulas given below and in A015442 in terms of S-Chebyshev polynomials are valid also for c^(-n), for n >= 0, with 1/c = (-1 + sqrt(29))/14 = A367454.

a(n) is the number of compositions (ordered partitions) of n into parts >= 2 and there are 7 sorts of each part. - Joerg Arndt, Jan 16 2024

Links

• Index entries for linear recurrences with constant coefficients, signature (1,7).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf.: A010484, A015442 (partial sums), A049310, A223140, A367454.

Programs

Mathematica

LinearRecurrence[{1,7},{1,0},30] (* James C. McMahon, Jan 16 2024 *)

Formula

a(n) = a(n-1) + 7*a(n-2), with a(0) = 1, a(1) = 0.

G.f.: (1 - x)/(1 - x - 7*x^2).

a(n) = 7*A015442(n-1), with A015442(-1) = 1/7.

a(n) = 7*(-i*sqrt(7))^(n-2)*S(n-2, i/sqrt(7)), with i = sqrt(-1) and the S-Chebyshev polynomial (see A049310). S(-2, x) = -1 and S(-1, x) = 0. The Fibonacci polynomials are F(n, x) = (-i)^(n-1)*S(n-1, i*x).