A078007 - OEIS

%I #24 Sep 08 2022 08:45:08

%S 1,0,2,3,7,15,32,69,148,318,683,1467,3151,6768,14537,31224,67066,

%T 144051,309407,664575,1427440,3065997,6585452,14144886,30381787,

%U 65257011,140165471,301061280,646649233,1388937264,2983297010,6407820771,13763352055,29562290607

%N Expansion of (1-x)/(1-x-2*x^2-x^3).

%C Let X = the 3x3 matrix [0,1,0; 0,0,1; 1,2,1]. a(n) = center term of X^n; but A002478(n) = term (3,3) of X^n. - _Gary W. Adamson_, May 30 2008

%C First bisection of A058278. - _Oboifeng Dira_, Aug 04 2016

%H G. C. Greubel, <a href="/A078007/b078007.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 (1,2,1).

%F a(n) = a(n-1) + 2*a(n-2) + a(n-3). - _Ilya Gutkovskiy_, Aug 06 2016

%t LinearRecurrence[{1,2,1}, {1,0,2}, 40] (* or *) CoefficientList[Series[(1 -x)/(1-x-2*x^2-x^3), {x,0,40}], x] (* _G. C. Greubel_, Jun 28 2019 *)

%o (PARI) Vec((1-x)/(1-x-2*x^2-x^3)+O(x^40)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-x-2*x^2-x^3) )); // _G. C. Greubel_, Jun 28 2019

%o (Sage) ((1-x)/(1-x-2*x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 28 2019

%o (GAP) a:=[1,0,2];; for n in [4..40] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Jun 28 2019

%Y First differences of A002478.

%Y Cf. A058278.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Nov 17 2002