

A203169


Sum of the fourth powers of the first n evenindexed Fibonacci numbers.


3



0, 1, 82, 4178, 198659, 9349284, 439330980, 20639983621, 969645224182, 45552722051318, 2140008541351943, 100534850436141384, 4722997973709689160, 221880369994471370761, 10423654392318557192602, 489689876072761951752602
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OFFSET

0,3


COMMENTS

Natural bilateral extension (brackets mark index 0): ..., 9349284, 198659, 4178, 82, 1, 0, [0], 1, 82, 4178, 198659, 9349284, ... That is, a(n) = a(n1).


LINKS

Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (56,440,770,440,56,1).


FORMULA

Let F(n) be the Fibonacci number A000045(n).
a(n) = sum_{k=1..n} F(2k)^4.
Closed form: a(n) = (1/75)(F(8n+4)  12 F(4n+2) + 9(2 n + 1)).
Recurrence: a(n)  56 a(n1) + 440 a(n2)  770 a(n3) + 440 a(n4)  56 a(n5) + a(n6) = 0.
G.f.: A(x) = (x + 26 x^2 + 26 x^3 + x^4)/(1  56 x + 440 x^2  770 x^3 + 440 x^4  56 x^5 + x^6) = x(1 + x)(1 + 25 x + x^2)/((1  x)^2 (1  7 x + x^2)(1  47 x + x^2)).


MATHEMATICA

a[n_Integer] := (1/75)(Fibonacci[8n+4]  12*Fibonacci[4n+2] + 9*(2*n+1)); Table[a[n], {n, 0, 20}]


CROSSREFS

Cf. A203170, A203171, A203172.
Cf. A027941, A103434, A163198.
Sequence in context: A035736 A017745 A217676 * A214815 A280959 A252705
Adjacent sequences: A203166 A203167 A203168 * A203170 A203171 A203172


KEYWORD

nonn,easy


AUTHOR

Stuart Clary, Dec 30 2011


STATUS

approved



