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A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4. 4
1, 4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n+1)/a(n) converges to 4 + sqrt(17).

LINKS

Table of n, a(n) for n=0..17.

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) = (((8+sqrt(68))/2)^n + ((8-sqrt(68))/2)^n)/2.

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

E.g.f.: exp(4*x)*cosh(sqrt(17)*x); a(n) = ((4+sqrt(17))^n+(4-sqrt(17))^n)/2; a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*17^k*4^(n-2*k). a(n) = T(n, 4*i)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003

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]

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

EXAMPLE

a(4) = 2177 = 8*a(3) + a(2) = 8*268 + 33 = (((8+sqrt(68))/2)^4 + ((8-sqrt(68))/2)^4)/2 = 2177.

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 */

CROSSREFS

Cf. A002018, A002190, A013192, A028576, A041027, A058153, A058155, A072754, A075132.

Cf. A041024.

Sequence in context: A081007 A213168 A203212 * A041024 A257068 A246806

Adjacent sequences:  A088314 A088315 A088316 * A088318 A088319 A088320

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Corrected generating function. - Philippe Deléham, Nov 20 2008

STATUS

approved

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Last modified November 20 00:42 EST 2017. Contains 294957 sequences.