login
Expansion of (1-x+x^2)*(1+x^4)/((1-x)^2*(1-x^2)).
1

%I #12 May 01 2023 08:36:54

%S 1,1,3,4,8,10,16,20,28,34,44,52,64,74,88,100,116,130,148,164,184,202,

%T 224,244,268,290,316,340,368,394,424,452,484,514,548,580,616,650,688,

%U 724,764,802,844,884,928,970,1016,1060,1108,1154,1204,1252,1304,1354,1408,1460

%N Expansion of (1-x+x^2)*(1+x^4)/((1-x)^2*(1-x^2)).

%C At one time this was given as the g.f. for A004657. In fact it produces a different sequence, showm here.

%H Colin Barker, <a href="/A092534/b092534.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F The g.f. can also be written as (1+x^2+x^4)*(1+x^4)/((1-x)*(1-x^2)*(1-x^3)). - _N. J. A. Sloane_, Nov 07 2017

%F From _Colin Barker_, Apr 14 2016: (Start)

%F a(n) = (13+3*(-1)^n-4*n+2*n^2)/4 for n>2.

%F a(n) = (n^2-2*n+8)/2 for n>2 and even.

%F a(n) = (n^2-2*n+5)/2 for n>2 and odd.

%F a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6) for n>6.

%F (End)

%t CoefficientList[Series[(1+x^2+x^4)(1+x^4)/((1-x)(1-x^2)(1-x^3)),{x,0,60}],x] (* or *) LinearRecurrence[{2,0,-2,1},{1,1,3,4,8,10,16},60] (* _Harvey P. Dale_, Nov 07 2017 *)

%o (PARI) Vec((1+x^2+x^4)*(1+x^4)/((1-x)*(1-x^2)*(1-x^3)) + O(x^50)) \\ _Colin Barker_, Apr 14 2016

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Apr 08 2004

%E Definition simplified by _N. J. A. Sloane_, Nov 07 2017