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A350901 a(n) = F(n) * (2*F(n-1)^2 + (-1)^(n-1)) * (2*F(n)^2 + (-1)^n), where F(n) is the n-th Fibonacci number.

%I A350901 #19 Jan 05 2025 19:51:42
%S A350901 0,1,3,42,399,4655,50568,565149,6248991,69380842,769112355,8530996299,
%T A350901 94604226192,1049202243593,11635724020011,129042610760010,
%U A350901 1431102560300007,15871178746661911,176014035001069464,1952025706821035013,21648296204009443815,240083286518079466826
%N A350901 a(n) = F(n) * (2*F(n-1)^2 + (-1)^(n-1)) * (2*F(n)^2 + (-1)^n), where F(n) is the n-th Fibonacci number.
%C A350901 This sequence is notable for having sum of reciprocals that is smaller by 2 than the sum of the reciprocals of the Fibonacci numbers (see the Formula section).
%C A350901 The series with the reciprocals of the positive terms of this sequence is converging much more rapidly. For example, to calculate the sum with an error that is smaller than 10^(-100) there is a need to sum up the reciprocals of the first 482 Fibonacci numbers, while with the reciprocals of the terms of this sequence the first 97 terms are enough.
%H A350901 Amiram Eldar, <a href="/A350901/b350901.txt">Table of n, a(n) for n = 0..300</a>
%H A350901 Richard André-Jeannin, <a href="https://fq.math.ca/Scanned/29-1/advanced29-1.pdf">Problem H-450</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 1 (1991), p. 89; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/30-2/advanced30-2.pdf">Comparable</a>, Solution to Problem H-450 by Paul S. Bruckman, ibid., Vol. 30, No. 2 (1992), pp. 191-192.
%H A350901 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,40,-60,-40,8,1).
%F A350901 a(n) = A000045(n) * A061646(n-1) * A061646(n).
%F A350901 a(n) = 8*a(n-1) + 40*a(n-2) - 60*a(n-3) - 40*a(n-4) + 8*a(n-5) + a(n-6), for n > 5.
%F A350901 Sum_{n>=1} 1/a(n) = -2 + A079586 (André-Jeannin, 1991).
%t A350901 a[n_] := Fibonacci[n]*(2*Fibonacci[n - 1]^2 + (-1)^(n - 1))*(2*Fibonacci[n]^2 + (-1)^n); Array[a, 25, 0]
%t A350901 (* or *)
%t A350901 LinearRecurrence[{8, 40, -60, -40, 8, 1}, {0, 1, 3, 42, 399, 4655}, 25]
%Y A350901 Cf. A000045, A061646, A079586.
%K A350901 nonn,easy
%O A350901 0,3
%A A350901 _Amiram Eldar_, Jan 21 2022