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<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2).
Limit(_{n->oo} a(n+1)/a(n), n=infinity) = 1+sqrt(2).
a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - _Colin Barker, _, Aug 29 2017
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(MAGMAMagma) I:=[1, 4, 10]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013
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From Clark Kimberling, Aug 23 2017 (Start)
p-INVERT of (1,1,1,....), where p(S) = 1-S-2*S^2+2*S^3.
Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453). See A291000 for a guide to related sequences.
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a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - Colin Barker, Aug 29 2017
(PARI) Vec((1 - 3*x^2) / ((1 - 2*x)*(1 - 2*x - x^2)) + O(x^30)) \\ Colin Barker, Aug 29 2017
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Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2*Pell(n+1)+2*Pell(n)-2^n, with P Pell = A000129.
G.f.: ( 1-3*x^2 ) / ( (1-42*x+3-1)*(x^2+2*x^3-1) ).
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