(MAGMAMagma) I:=[3, 2, 7, 12]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 24 2021
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Expansion of (-3+ -4*x+ -3*x^2)/((1-x^2)*(x+1)*(x^2+-2*x-1x^2)); a Pellian-related sequence.
G.f.: (3 -4*x -3*x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
a(n) = A000129(n+1) + 2*A059841(n). - R. J. Mathar, Nov 10 2009
a(n) = (1 + (-1)^n + (-(1-+sqrt(2))^(1+n)+ - (1+-sqrt(2))^(1+n))/(2*sqrt(2))).
a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) for n>3. (End)
(End)
a(n) = A000129(n+1) + 1 + (-1)^n. - G. C. Greubel, May 24 2021
Table[Fibonacci[n+1, 2] +1+(-1)^n, {n, 0, 30}] (* G. C. Greubel, May 24 2021 *)
(PARI) Vec((-3+-4*x+-3*x^2)/((1-x^2)*(x+1)*(x^2+-2*x-1x^2)) + O(x^50)) \\ Colin Barker, May 26 2016
(MAGMA) I:=[3, 2, 7, 12]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 24 2021
(Sage) [lucas_number1(n+1, 2, -1) +(1+(-1)^n) for n in (0..30)] # G. C. Greubel, May 24 2021
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3, 2, 7, 12, 31, 70, 171, 408, 987, 2378, 5743, 13860, 33463, 80782, 195027, 470832, 1136691, 2744210, 6625111, 15994428, 38613967, 93222358, 225058683, 543339720, 1311738123, 3166815962, 7645370047, 18457556052, 44560482151, 107578520350, 259717522851
Colin Barker, <a href="/A114647/b114647.txt">Table of n, a(n) for n = 0..1000</a>
<a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
a(n) = A000129(n+1)+2*A059841(n). [From _- _R. J. Mathar_, Nov 10 2009]
From Colin Barker, May 26 2016: (Start)
a(n) = (1+(-1)^n+(-(1-sqrt(2))^(1+n)+(1+sqrt(2))^(1+n))/(2*sqrt(2))).
a(n) = 2*a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4) for n>3.
(End)
(PARI) Vec((-3+4*x+3*x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, May 26 2016
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_Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), _, Feb 18 2006