A243006 - OEIS

%I #15 Sep 08 2022 08:46:08

%S 0,1,-1,7,27,25,2,13,54,49,5,19,81,73,8,25,108,97,11,31,135,121,14,37,

%T 162,145,17,43,189,169,20,49,216,193,23,55,243,217,26,61,270,241,29,

%U 67,297,265,32,73,324,289,35,79,351,313,38,85,378,337,41,91,405

%N G.f.: (x - 3*x^2 + 12*x^3 + 6*x^4 - x^5) / ((1 - x) * (1 + x^2))^2.

%H G. C. Greubel, <a href="/A243006/b243006.txt">Table of n, a(n) for n = 0..2500</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,4,-3,2,-1)

%F a(4*n) = 27*n, a(4*n + 1) = 24*n + 1, a(4*n + 2) = 3*n - 1, a(4*n - 1) = 6*n + 1.

%F 0 = a(n)*(+a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-2*a(n+1) + 3*a(n+2) + 2*a(n+3)) + a(n+2)*(-2*a(n+2) + a(n+3)) for all n in Z.

%e G.f. = x - x^2 + 7*x^3 + 27*x^4 + 25*x^5 + 2*x^6 + 13*x^7 + 54*x^8 + ...

%t CoefficientList[Series[(x-3*x^2+12*x^3+6*x^4-x^5)/((1-x)*(1+x^2))^2, {x, 0, 60}], x]  (* _G. C. Greubel_, Aug 06 2018 *)

%o (PARI) {a(n) = my(m = n\4); [27*m, 24*m + 1, 3*m - 1, 6*m + 7][n%4 + 1]};

%o (PARI) {a(n) = polcoeff( if( n<0, (x - 6*x^2 - 12*x^3 + 3*x^4 - x^5), (x - 3*x^2 + 12*x^3 + 6*x^4 - x^5)) / ((1 - x) * (1 + x^2))^2 + x * O(x^abs(n)), abs(n))};

%o (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((x-3*x^2+12*x^3+6*x^4-x^5)/((1-x)*(1+x^2))^2)); // _G. C. Greubel_, Aug 06 2018

%K sign,easy

%O 0,4

%A _Michael Somos_, Aug 17 2014