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A264080 a(n) = 6*F(n)*F(n+1) + (-1)^n, where F = A000045.

%I A264080 #48 Sep 08 2022 08:46:14
%S A264080 1,5,13,35,91,239,625,1637,4285,11219,29371,76895,201313,527045,
%T A264080 1379821,3612419,9457435,24759887,64822225,169706789,444298141,
%U A264080 1163187635,3045264763,7972606655,20872555201,54645058949,143062621645,374542805987,980565796315
%N A264080 a(n) = 6*F(n)*F(n+1) + (-1)^n, where F = A000045.
%C A264080 a(n) is prime for n = 1, 2, 5, 7, 14, 15, 29, 40, 49, 57, 70, 87, 105, 127, 175, 279, 362, 647, 727, ...
%H A264080 Bruno Berselli, <a href="/A264080/b264080.txt">Table of n, a(n) for n = 0..500</a>
%H A264080 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A264080 G.f.: (1+3*x+x^2) / ((1+x)*(1-3*x+x^2)). - Corrected by _Colin Barker_, Sep 28 2016
%F A264080 a(n) = -a(-n-1) = 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z.
%F A264080 a(n) = L(2*n+1) + F(n)*F(n+1) = A002878(n) + A001654(n). See similar identity for A061647.
%F A264080 a(n) = A001654(n+1) + 3*A001654(n) + A001654(n-1).
%F A264080 a(n) - a(n-1) = 2*A099016(n) with a(-1)=-1.
%F A264080 a(n) + a(n-1) = 2*A097134(n) for n>0.
%F A264080 Sum_{i>=0} 1/a(i) = 1.3232560865206157372628688449331...
%F A264080 a(n) = (2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5. - _Colin Barker_, Sep 28 2016
%F A264080 E.g.f.: (1/5)*exp(-x)*(-1 + 6*exp(5*x/2)*(cosh((sqrt(5)*x)/2) + sqrt(5)*sinh((sqrt(5)*x)/2))). - _Stefano Spezia_, Dec 09 2019
%p A264080 a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,5,13>>)[1, 1]:
%p A264080 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 28 2016
%t A264080 Table[6 Fibonacci[n] Fibonacci[n + 1] + (-1)^n, {n, 0, 30}]
%t A264080 LinearRecurrence[{2,2,-1},{1,5,13},30] (* _Harvey P. Dale_, Jul 12 2019 *)
%o A264080 (Sage) [6*fibonacci(n)*fibonacci(n+1)+(-1)^n for n in (0..30)]
%o A264080 (Maxima) makelist(6*fib(n)*fib(n+1)+(-1)^n, n, 0, 30);
%o A264080 (Magma) [6*Fibonacci(n)*Fibonacci(n+1)+(-1)^n: n in [0..30]];
%o A264080 (PARI) for(n=0, 30, print1(6*fibonacci(n)*fibonacci(n+1)+(-1)^n", "));
%o A264080 (PARI) a(n) = round((2^(-n)*(-(-2)^n-3*(3-sqrt(5))^n*(-1+sqrt(5))+3*(1+sqrt(5))*(3+sqrt(5))^n))/5) \\ _Colin Barker_, Sep 28 2016
%o A264080 (PARI) Vec((1+3*x+x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Sep 28 2016
%Y A264080 Cf. A000032, A000045; A001654, A002878.
%Y A264080 Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n: A226205 (k=1); A236428 (k=2); A014742 (k=3); A061647 (k=4); A002878 (k=5).
%K A264080 nonn,easy
%O A264080 0,2
%A A264080 _Bruno Berselli_, Nov 03 2015