A140413 - OEIS

%I #38 Nov 03 2025 10:47:05

%S 1,1,9,33,145,609,2585,10945,46369,196417,832041,3524577,14930353,

%T 63245985,267914297,1134903169,4807526977,20365011073,86267571273,

%U 365435296161,1548008755921,6557470319841,27777890035289,117669030460993,498454011879265,2111485077978049

%N a(2n) = A000045(6n) + 1, a(2n+1) = A000045(6n+3) - 1.

%H Colin Barker, <a href="/A140413/b140413.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,5,1).

%F a(n) = A141325(3*n) = (-1)^n + A014445(n).

%F From _R. J. Mathar_, Dec 17 2010: (Start)

%F a(n) = +3*a(n-1) +5*a(n-2) +a(n-3).

%F G.f.: (1-x)^2 / ( (1+x)*(1-4*x-x^2) ). (End)

%F a(n) = ((-1)^n + (-(2-sqrt(5))^n + (2+sqrt(5))^n) / sqrt(5)). - _Colin Barker_, Jun 06 2017

%F a(n) = -A033887(n) + 2*Sum_{k=0..n} A033887(k)*(-1)^(n-k). - Yomna Bakr and _Greg Dresden_, Jun 03 2024

%F E.g.f.: cosh(x) - sinh(x) + 2*exp(2*x)*sinh(sqrt(5)*x)/sqrt(5). - _Stefano Spezia_, Nov 03 2025

%t LinearRecurrence[{3,5,1},{1,1,9},30] (* or *) CoefficientList[Series[ (1-x)^2/((1+x)(1-4*x-x^2)),{x,0,30}],x] (* _Harvey P. Dale_, Jun 20 2011 *)

%o (PARI) Vec((1-x)^2/((1+x)*(1-4*x-x^2)) + O(x^30)) \\ _Colin Barker_, Jun 06 2017

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)^2/((1+x)*(1-4*x-x^2)) )); // _G. C. Greubel_, Jun 08 2019

%o (SageMath) ((1-x)^2/((1+x)*(1-4*x-x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 08 2019

%o (GAP) a:=[1,1,9];; for n in [4..30] do a[n]:=3*a[n-1]+5*a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Jun 08 2019

%Y Cf. A000045, A014445, A033887, A141325.

%K nonn,easy

%O 0,3

%A _Paul Curtz_, Jun 17 2008