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A132357 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,4,14,41.

%I A132357 #30 Feb 03 2025 18:56:14
%S A132357 1,4,14,41,122,364,1093,3280,9842,29525,88574,265720,797161,2391484,
%T A132357 7174454,21523361,64570082,193710244,581130733,1743392200,5230176602,
%U A132357 15690529805,47071589414,141214768240,423644304721
%N A132357 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,4,14,41.
%H A132357 Paolo Xausa, <a href="/A132357/b132357.txt">Table of n, a(n) for n = 0..1000</a>
%H A132357 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-1,3).
%F A132357 O.g.f.: -(1+x+2*x^2)/((3*x-1)*(x+1)*(x^2-x+1)) = -(3/2)/(3*x-1)+(1/3)*(x-2)/(x^2-x+1)+(1/ 6)/(x+1). - _R. J. Mathar_, Nov 28 2007
%F A132357 a(n) = (1/2)*3^(n+1) + (1/6)*(-1)^n - (2/3)*cos(Pi*n/3). Or, a(n) = (1/2)*3^(n+1) + (1/2)*[ -1; -1; 1; 1; 1; -1]. - _Richard Choulet_, Jan 02 2008
%F A132357 a(n+1) - 3a(n) = A132367(n+1). - _Paul Curtz_, Dec 02 2007
%F A132357 6*a(n) = (-1)^n +3^(n+2) -2*A057079(n+1). - _R. J. Mathar_, Oct 03 2021
%t A132357 LinearRecurrence[{3,0,-1,3},{1,4,14,41},50] (* _Paolo Xausa_, Dec 05 2023 *)
%o A132357 (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 3,-1,0,3]^n*[1;4;14;41])[1,1] \\ _Charles R Greathouse IV_, Oct 08 2016
%Y A132357 First differences of A132353.
%Y A132357 Cf. A129339.
%K A132357 nonn,easy
%O A132357 0,2
%A A132357 _Paul Curtz_, Nov 24 2007