A365824 a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.

1, 0, 5, 5, 30, 55, 205, 480, 1505, 3905, 11430, 30955, 88105, 242880, 683405, 1897805, 5314830, 14803855, 41378005, 115397280, 322287305, 899273705, 2510710230, 7007078755, 19560629905, 54596023680, 152399173205, 425379291605, 1187375157630, 3314271615655

Offset: 0

Author

Wolfdieter Lang, Nov 20 2023

Keywords

• nonn
• easy

Comments

This sequence {a(n)} appears in the formula for powers of phi21 := (1 + sqrt(21))/2 = A222134 = 2.791287..., together with b(n) = A015440(n-1), with A015440(-1) = 0, as phi21^n = a(n) + b(n)*phi21(n), for n >= 0. But the later given formulas in terms of scaled Chebyshev polynomials, called here {S21(n)}, are valid also for negative n values, i.e., for nonnegative powers of 1/phi21 = (-1 + sqrt(21))/10 = 0.35825756949... = A367453.

Limit_{n->oo} a(n)/a(n-1) = (1 + sqrt(21))/2 = A222134 = 2.791287...

Examples

			phi21^2 = a(2) + b(2)*phi(n) = 5 + phi21 = 7.79128784..., a real algebraic integer in Q(sqrt(21)).
(1/phi21)^2 = a(-2) + b(-2)*phi21 = (1/25)*(6 - phi21) = 0.12834848..., a real algebraic number in Q(sqrt(21)).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,5).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A010477 (sqrt(21)), A015440, A049310, A222134, A367453.

Programs

Mathematica

LinearRecurrence[{1,5},{1,0},50] (* Paolo Xausa, Nov 21 2023 *)

PARI

a(n) = abs([1, 3; 1, -2]^(n-2)*[5; 5])[2, 1] \\ Thomas Scheuerle, Nov 20 2023

Formula

a(n) = a(n-1) + 5*a(n-2), for n >= 0, with a(0) = 1 and a(1) = 0.

G.f.: (1 - x)/(1 - x - 5*x^2).

a(n) = S21(n+1) - S21(n), for n >= 0, where S21(n) = sqrt(-5)^(n-1)*S(n-1, 1/sqrt(-5)), with the Chebyshev polynomials {S(n, x)} (see A049310).

The above mentioned sequence {b(n)} has terms b(n) = A015440(n-1) = S21(n), for n >= 0, with the same recurrence as {a(n)} but with b(0) = 0 and b(1) = 1, and g.f. x/(1 - x - 5*x^2).

The formula for negative indices of S is: S(-1, 0) = 0 and S(-n, x) = -S(n-2, x) for n >= 2.