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A049668 a(n) = F(8n)/21, where F = A000045 (the Fibonacci sequence). 8
0, 1, 47, 2208, 103729, 4873055, 228929856, 10754830177, 505248088463, 23735905327584, 1115082302307985, 52385132303147711, 2460986135945634432, 115613963257141670593, 5431395286949712883439, 255159964523379363851040, 11987086937311880388115441 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Colin Barker, Table of n, a(n) for n = 0..500

R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (47,-1).

FORMULA

G.f.: x/(1-47*x+x^2), 47=L(8)=A000032(8) (Lucas).

a(n) = 47*a(n-1)-a(n-2) ; a(0)=0, a(1)=1. - Philippe Deléham, Nov 18 2008

From Peter Bala, Apr 03 2015: (Start)

For integer k, 1 + k*(14 - k)*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + k/3*Sum_{n >= 1} Fibonacci(4*n)*x^n )*( 1 + k/3*Sum_{n >= 1} Fibonacci(4*n)*(-x)^n ).

1 + 45*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + Sum_{n >= 1} Lucas(4*n)*x^n )*( 1 + Sum_{n >= 1} Lucas(4*n)*(-x)^n ).

1 - 36*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + 2*Sum_{n >= 1} Fibonacci(4*n+2)*x^n )*( 1 + 2*Sum_{n >= 1} Fibonacci(4*n+2)*(-x)^n ). (End)

a(n) = ((47+21*sqrt(5))^(1-n)*(-2^n+(2207+987*sqrt(5))^n))/(2205+987*sqrt(5)). - Colin Barker, Jun 03 2016

MATHEMATICA

Table[Fibonacci[8 n]/21, {n, 15}] (* Michael De Vlieger, Apr 03 2015 *)

PROG

(Mupad) numlib::fibonacci(8*n)/21 $ n = 0..25; # Zerinvary Lajos, May 09 2008

(PARI) concat(0, Vec(x/(1-47*x+x^2) + O(x^20))) \\ Colin Barker, Jun 03 2016

CROSSREFS

A column of array A028412.

Sequence in context: A170728 A170766 A218749 * A009991 A052463 A005148

Adjacent sequences:  A049665 A049666 A049667 * A049669 A049670 A049671

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 28 05:37 EDT 2017. Contains 285557 sequences.