A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

1, 4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236, 782369683393860737, 6355271576489378132, 51624542295308885793

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (8,1).

Formula

a(n) = ( (4+sqrt(17))^n + (4-sqrt(17))^n )/2.

a(n) = A086594(n)/2.

Lim_{n -> oo} a(n+1)/a(n) = 4 + sqrt(17).

From Paul Barry, Nov 15 2003: (Start)

E.g.f.: exp(4*x)*cosh(sqrt(17)*x).

a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*17^k*4^(n-2*k).

a(n) = (-i)^n * T(n, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)

a(n) = A041024(n-1), n>0. - R. J. Mathar, Sep 11 2008

G.f.: (1-4*x)/(1-8*x-x^2). - Philippe Deléham, Nov 16 2008 and Nov 20 2008

a(n) = (1/2)*((33+8*sqrt(17))*(4-sqrt(17))^(n+2) + (33-8*sqrt(17))*(4+sqrt(17))^(n+2)). - Harvey P. Dale, May 07 2012

Mathematica

LinearRecurrence[{8, 1}, {1, 4}, 30] (* or *) With[{c=Sqrt[17]}, Simplify/@ Table[1/2 (c-4)((c+4)^n-(4-c)^n (33+8c)), {n, 30}]] (* Harvey P. Dale, May 07 2012 *)

Prog

(Maxima)
a[0]:1$ a[1]:4$ a[n]:=8*a[n-1]+a[n-2]$ A088317(n):=a[n]$
makelist(A088317(n), n, 0, 20); /* Martin Ettl, Nov 12 2012 */
(Magma) [n le 2 select 4^(n-1) else 8*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022
(SageMath)
A088317=BinaryRecurrenceSequence(8, 1, 1, 4)
[A088317(n) for n in range(31)] # G. C. Greubel, Dec 13 2022

Crossrefs

Cf. A002018, A002190, A013192, A028576, A041024, A041027.

Cf. A053120, A058153, A058155, A072754, A075132.

Sequence in context: A213168 A203212 A041024 * A257068 A246806 A202765

Adjacent sequences: A088314 A088315 A088316 * A088318 A088319 A088320

Keyword

nonn,easy

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003

Status

approved