A128698 - OEIS

%I #15 Sep 08 2022 08:45:30

%S 0,-1,0,-256,6305,-384320,16392896,-799337825,37023521536,

%T -1748770383360,81985167507265,-3854603638194816,181029655256841600,

%U -8505521232849819841,399560845889490455040,-18771170453838609544960,881839776158402870049761,-41427800130507702988683200,1946222939243803281837279296,-91431083130550578762727373345,4295314095871701743501398017280

%N Alternating sum of the eighth powers of the first n Fibonacci numbers.

%C Natural bilateral extension (brackets mark index 0): ..., -16392896, 384320, -6305, 256, 0, 1, 0, [0], -1, 0, -256, 6305, -384320, 16392896, ... This is (-A128698)-reversed followed by A128698.

%H G. C. Greubel, <a href="/A128698/b128698.txt">Table of n, a(n) for n = 0..595</a>

%F Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).

%F a(n) = Sum_{k=1..n} (-1)^k F(k)^8.

%F Closed form: a(n) = (-1)^n L(8n+4)/4375 - 2 L(6n+3)/625 + (-1)^n 28 L(4n+2)/1875 - 56 L(2n+1)/625 + (-1)^n 7/125.

%F Factored closed form: a(n) = (-1)^n (1/21) F(n-2) F(n) F(n+1) F(n+3) (3 F(n)^2 F(n+1)^2 + 4).

%F Recurrence: a(n) + 34 a(n-1) - 714 a(n-2) - 4641 a(n-3) + 12376 a(n-4) + 12376 a(n-5) - 4641 a(n-6) - 714 a(n-7) + 34 a(n-8) + a(n-9) = 0.

%F G.f.: A(x) = (-x - 34 x^2 + 458 x^3 + 2242 x^4 + 458 x^5 - 34 x^6 - x^7)/(1 + 34 x - 714 x^2 - 4641 x^3 + 12376 x^4 + 12376 x^5 - 4641 x^6 - 714 x^7 + 34 x^8 + x^9) = -x(1 + 34 x - 458 x^2 - 2242 x^3 - 458 x^4 + 34 x^5 + x^6)/((1 + x)(1 - 3 x + x^2)(1 + 7 x + x^2)(1 - 18 x + x^2)(1 + 47 x + x^2)).

%t a[ n_Integer ] := If[ n >= 0, Sum[ (-1)^k Fibonacci[ k ]^8, {k, 1, n} ], Sum[ -(-1)^k Fibonacci[ -k ]^8, {k, 1, -n - 1} ] ]

%t Accumulate[Times@@@Partition[Riffle[Fibonacci[Range[0,20]]^8,{1,-1},{2,-1,2}],2]] (* _Harvey P. Dale_, May 04 2016 *)

%o (PARI) a(n) = sum(k=1, n, (-1)^k*fibonacci(k)^8); \\ _Michel Marcus_, Dec 10 2016

%o (Magma) [(&+[(-1)^k*Fibonacci(k)^8: k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jan 17 2018

%Y Cf. A128697, A119282, A119283, A119284, A119285, A119286, A119287, A128696.

%K sign,easy

%O 0,4

%A _Stuart Clary_, Mar 23 2007