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%I #37 Sep 08 2022 08:46:13
%S -1,2,1,7,14,41,103,274,713,1871,4894,12817,33551,87842,229969,602071,
%T 1576238,4126649,10803703,28284466,74049689,193864607,507544126,
%U 1328767777,3478759199,9107509826,23843770273,62423800999,163427632718,427859097161,1120149658759
%N a(n) = F(n)*F(n+1) - (-1)^n, where F = A000045.
%C Primes in sequence for n = 1, 3, 5, 6, 9, 24, 42, 48, 53, 71, 86, 102, 138, 182, 302, 438, 506, 926, ...
%H Bruno Berselli, <a href="/A260259/b260259.txt">Table of n, a(n) for n = 0..500</a>
%H A. Bremner, R. Høibakk, D. Lukkassen, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_36_from29to41.pdf">Crossed ladders and Euler’s quartic</a>, Annales Mathematicae et Informaticae, 36 (2009) pp. 29-41. See p. 33.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F G.f.: (-1 + 4*x - x^2)/((1 + x)*(1 - 3*x + x^2)).
%F a(n) = -a(-n-1) = 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z.
%F a(n) = F(n+2)^2 - 2*F(n+1)^2.
%F a(n) = A059929(n) - A059929(n-1) with A059929(-1)=1.
%F a(n) = -A001654(n+1) + 4*A001654(n) - A001654(n-1).
%F a(n) = A206351((n+2)/2)-2 for even n; a(n) = A003482((n-1)/2)+2 for odd n.
%F Sum_{i>=0} 1/a(i) = .754301907697893871765121109686...
%F a(n) = (2^(-1-n)*(-3*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/5. - _Colin Barker_, Sep 29 2016
%p with(combinat): A260259:=n->fibonacci(n)*fibonacci(n+1)-(-1)^n: seq(A260259(n), n=0..50); # _Wesley Ivan Hurt_, Feb 04 2017
%t Table[Fibonacci[n] Fibonacci[n + 1] - (-1)^n, {n, 0, 30}]
%o (PARI) for(n=0, 30, print1(fibonacci(n)*fibonacci(n+1)-(-1)^n", "));
%o (PARI) a(n) = round((2^(-1-n)*(-3*(-1)^n*2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ _Colin Barker_, Sep 29 2016
%o (PARI) Vec(-(1-4*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Sep 29 2016
%o (Sage) [fibonacci(n)*fibonacci(n+1)-(-1)^n for n in (0..30)]
%o (Maxima) makelist(fib(n)*fib(n+1)-(-1)^n,n,0,30);
%o (Magma) [Fibonacci(n)*Fibonacci(n+1)-(-1)^n: n in [0..30]];
%Y First bisection of A111569.
%Y Cf. A226205: numbers of the form F(n)*F(n+1)+(-1)^n.
%Y Cf. A000045, A001654, A003482, A059929, A089508 (first bisection, without -1), A206351.
%K sign,easy
%O 0,2
%A _Bruno Berselli_, Oct 31 2015
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