cp's OEIS Frontend

A114695 Three consecutive elements of the sequence built from a quadratic form over four consecutive Fibonacci numbers A000045.

%I A114695 #8 May 25 2021 01:37:39
%S A114695 2,2,4,104,143,169,4895,6764,7921,229970,317810,372100,10803704,
%T A114695 14930351,17480761,507544127,701408732,821223649,23843770274,
%U A114695 32951280098,38580030724,1120149658760,1548008755919,1812440220361
%N A114695 Three consecutive elements of the sequence built from a quadratic form over four consecutive Fibonacci numbers A000045.
%H A114695 G. C. Greubel, <a href="/A114695/b114695.txt">Table of n, a(n) for n = 0..500</a>
%H A114695 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,48,0,0,-48,0,0,1).
%F A114695 a(3*n) = Fibonacci(4*n+2)*Fibonacci(4*n+3).
%F A114695 a(3*n+1) = Lucas(4*n+1)*Fibonacci(4*n+3).
%F A114695 a(3*n+2) = Fibonacci(4*n+3)*Fibonacci(4*n+3).
%F A114695 From _R. J. Mathar_, Apr 16 2009: (Start)
%F A114695 a(3*n) = A001654(4*n+2).
%F A114695 a(3*n+1) = A128535(4*n+3).
%F A114695 a(3*n+2) = A007598(4*n+3).
%F A114695 G.f.: (2+2*x+4*x^2+8*x^3+47*x^4-23*x^5-x^6-4*x^7+x^8)/((1-x)*(1+x+x^2)*(1-47*x^3+x^6)).
%F A114695 a(n) = 48*a(n-3) - 48*a(n-6) + a(n-9). (End)
%F A114695 a(n) = F(4*m+3)*( 4*F(4*m+2) - ((n^2 mod 3)*F(4*m+2) + ((n+2)^2 mod 3)*Lucas(4*m+1) + ((n+1)^2 mod 3)*F(4*m+3)) ), where m = floor(n/3) and F = Fibonacci. - _G. C. Greubel_, May 24 2021
%t A114695 F[n_]:= Fibonacci[n]; Flatten[Table[{F[4*n+2]*F[4*n+3], (F[4*n]+F[4*n+2])*F[4*n+ 3], F[4*n+3]^2}, {n, 0, 12}]] (* modified by _G. C. Greubel_, May 24 2021 *)
%t A114695 With[{m = Floor[n/3], F = Fibonacci}, Table[F[4*m+3]*(4*F[4*m+2] -(Mod[n^2,3]*F[4*m +2] +Mod[(n+2)^2, 3]*LucasL[4*m+1] +Mod[(n+1)^2, 3]*F[4*m+3])), {n, 0, 40}]] (* _G. C. Greubel_, May 24 2021 *)
%o A114695 (Sage)
%o A114695 f=fibonacci;
%o A114695 def A114695(n): return f(4*(n//3)+3)*( 4*f(4*(n//3)+2) - ((n^2%3)*f(4*(n//3)+2) + ((n+2)^2%3)*(f(4*(n//3)+2) + f(4*(n//3))) + ((n+1)^2%3)*f(4*(n//3)+3) ) )
%o A114695 [A114695(n) for n in (0..40)] # _G. C. Greubel_, May 24 2021
%Y A114695 Cf. A000032, A000045.
%K A114695 nonn,less
%O A114695 0,1
%A A114695 _Roger L. Bagula_, Feb 21 2006
%E A114695 Edited by the Associate Editors of the OEIS, Sep 02 2009