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%I #25 Sep 21 2023 11:35:43
%S 2,2,1,7,3,16,12,36,39,85,113,210,310,534,829,1379,2191,3588,5760,
%T 9368,15107,24497,39581,64102,103658,167786,271417,439231,710619,
%U 1149880,1860468,3010380,4870815,7881229,12752009,20633274,33385246,54018558,87403765
%N Sequence derived from arithmetic relations between powers of phi (A001622): a(n) = phi^n - (-1)^n * (n - phi^-n).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PhiNumberSystem.html">Phi Number System</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,2,3,1).
%F a(n) = phi^n - (-1)^n * (n - phi^-n), phi = (1 + sqrt(5))/2 = A001622.
%F G.f.: (2*x+1)*(x^2-2)/((x^2+x-1)*(x+1)^2). - _Alois P. Heinz_, Oct 17 2014
%F a(n) = A000032(n) - (-1)^n*n. - _Alois P. Heinz_, Oct 17 2014
%e a(7) = phi^7 + (n - phi^-7) = 36; a(10) = phi^10 - (n - phi^-10) = 113.
%t LinearRecurrence[{-1,2,3,1},{2,2,1,7},40] (* _Harvey P. Dale_, Sep 21 2023 *)
%o (PARI) a(n)=fibonacci(n-1) + fibonacci(n+1) - n*(-1)^n \\ _Charles R Greathouse IV_, Oct 28 2014
%Y Cf. A000032, A001622.
%K nonn,easy
%O 0,1
%A _Gustavo Mendoza_, Oct 16 2014
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