Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #11 Aug 30 2024 10:00:54
%S 12,40,1365,19448,381276,6615103,120241980,2147070680,38600066517,
%T 692153278024,12423591148332,222908960952575,4000098954110700,
%U 71777766990248968,1288007282149222101,23112301389881302808,414733773612913239420,7442093184423393874495,133542960264663589170972
%N a(n) = Lucas(n-1) * Lucas(n+1) * Fibonacci(2*n-1) * Fibonacci(2*n+1).
%H Amiram Eldar, <a href="/A375804/b375804.txt">Table of n, a(n) for n = 1..798</a>
%H Hideyuki Ohtskua, proposer, <a href="https://www.fq.math.ca/Problems/AugAdvProb2024.pdf">Problem H-944</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 3 (2024), p. 266.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (13,104,-260,-260,104,13,-1).
%F a(n) = A292696(n) * A064170(n+2).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(5) - 2)/ 4 = A204188 - 1/2 (Ohtskua, 2024).
%F G.f.: -x^2*(-20+65*x+195*x^2-84*x^3-13*x^4+x^5)/ ( (1+x) *(x^2-3*x+1) *(x^2+7*x+1) *(x^2-18*x+1) ). - _R. J. Mathar_, Aug 30 2024
%t a[n_] := LucasL[n-1] * LucasL[n+1] * Fibonacci[2*n-1] * Fibonacci[2*n+1]; Array[a, 20]
%o (PARI) lucas(n) = fibonacci(n-1) + fibonacci(n+1);
%o a(n) = lucas(n-1) * lucas(n+1) * fibonacci(2*n-1) * fibonacci(2*n+1);
%Y Cf. A000032, A000045, A064170, A204188, A292696, A375803.
%K nonn,easy
%O 1,1
%A _Amiram Eldar_, Aug 29 2024