%I #20 Sep 08 2022 08:45:30

%S 0,-1,0,-128,2059,-76066,2021086,-60727431,1740361110,-50782989034,

%T 1471652245341,-42759682650188,1241158781898676,-36040175501820901,

%U 1046363981321362852,-30381064378888637148,882092032492683277335,-25611107658594421205278,743603574761804566730466,-21590121866471006254739195,626857059065125789349713930

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

%C Natural bilateral extension (brackets mark index 0): ..., 2177594, 80442, 2317, 130, 2, 1, 0, [0], -1, 0, -128, 2059, -76066, 2021086 ... This is A098533-reversed followed by A128696.

%H G. C. Greubel, <a href="/A128696/b128696.txt">Table of n, a(n) for n = 0..680</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (-20,294,819,-2912,728,1365,-252,-22,1).

%F Let F(n) be the Fibonacci number A000045(n).

%F a(n) = Sum_{k=1..n} (-1)^k F(k)^7.

%F Closed form: a(n) = (-1)^n (F(7n+7) - F(7n))/3625 + 7(F(5n+1) - 2 F(5n+4))/1375 + (-1)^n 21 F(3n+1)/250 - 7 F(n+2)/25 + 139/638.

%F Recurrence: a(n) + 20 a(n-1) - 294 a(n-2) - 819 a(n-3) + 2912 a(n-4) - 728 a(n-5) - 1365 a(n-6) + 252 a(n-7) + 22 a(n-8) - a(n-9) = 0.

%F G.f.: A(x) = (-x - 20 x^2 + 166 x^3 + 318 x^4 - 166 x^5 - 20 x^6 + x^7)/(1 + 20 x - 294 x^2 - 819 x^3 + 2912 x^4 - 728 x^5 - 1365 x^6 + 252 x^7 + 22 x^8 - x^9) = -x*(1 + 20 x - 166 x^2 - 318 x^3 + 166 x^4 + 20 x^5 - x^6)/ ((1 - x)*(1 - x - x^2)*(1 + 4 x - x^2)*(1 - 11 x - x^2)*(1 + 29 x - x^2)).

%t a[ n_Integer ] := If[ n >= 0, Sum[ (-1)^k Fibonacci[ k ]^7, {k, 1, n} ], Sum[ -(-1)^k Fibonacci[ -k ]^7, {k, 1, -n - 1} ] ]

%t Accumulate[Times@@@Partition[Riffle[Fibonacci[Range[0,30]]^7,{1,-1}], 2]] (* _Harvey P. Dale_, May 11 2012 *)

%o (PARI) a(n) = sum(k=1, n, (-1)^k*fibonacci(k)^7); \\ _Michel Marcus_, Dec 10 2016

%o (Magma) [(&+[(-1)^k*Fibonacci(k)^7: k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jan 17 2018

%Y Cf. A098533, A119282, A119283, A119284, A119285, A119286, A119287, A128698.

%K sign,easy

%O 0,4

%A _Stuart Clary_, Mar 23 2007