cp's OEIS Frontend

A203170 Sum of the fourth powers of the first n odd-indexed Fibonacci numbers.

%I A203170 #12 Jun 13 2025 11:33:38
%S A203170 0,1,17,642,29203,1365539,64107780,3011403301,141469813301,
%T A203170 6646055880582,312223061019703,14667837157106759,689076118833981960,
%U A203170 32371909717271872585,1520790680382055836761,71444790066793903279242
%N A203170 Sum of the fourth powers of the first n odd-indexed Fibonacci numbers.
%C A203170 Natural bilateral extension (brackets mark index 0): ..., -1365539, -29203, -642, -17, -1, [0], 1, 17, 642, 29203, 1365539, ...  That is, a(-n) = -a(n).
%H A203170 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (56,-440,770,-440,56,-1).
%F A203170 Let F(n) be the Fibonacci number A000045(n).
%F A203170 a(n) = Sum_{k=1..n} F(2k-1)^4.
%F A203170 Closed form: a(n) = (1/75)(F(8n) + 12 F(4n) + 18 n).
%F A203170 Recurrence: a(n) - 56 a(n-1) + 440 a(n-2) - 770 a(n-3) + 440 a(n-4) - 56 a(n-5) + a(n-6) = 0.
%F A203170 G.f.: A(x) = (x - 39 x^2 + 130 x^3 - 39 x^4 + x^5)/(1 - 56 x + 440 x^2 - 770 x^3 + 440 x^4 - 56 x^5 + x^6) = x(1 - 39 x + 130 x^2 - 39 x^3 + x^4)/((1 - x)^2 (1 - 7 x + x^2)(1 - 47 x + x^2)).
%t A203170 a[n_Integer] := (1/75)(Fibonacci[8n] + 12*Fibonacci[4n] + 18 n); Table[a[n], {n, 0, 20}]
%Y A203170 Cf. A203169, A203171, A203172.
%Y A203170 Cf. A001906, A103433, A163200.
%K A203170 nonn,easy
%O A203170 0,3
%A A203170 _Stuart Clary_, Dec 30 2011