|
%I #31 Sep 08 2022 08:44:49
%S 1,3,7,19,47,123,311,803,2047,5259,13447,34483,88271,226203,579287,
%T 1484099,3801247,9737643,24942631,63893203,163663727,419236539,
%U 1073891447,2750837603,7046403391,18049753803,46235367367,118434382579,303375852047,777113382363
%N Expansion of (1 + 2*x) / (1 - x - 4*x^2).
%C T(n,0) + T(n,1) + ... + T(n,2n), T given by A026568.
%C Row sums of Riordan array ((1+2x)/(1+x),x(1+2x)/(1+x)). Binomial transform is A055099. - _Paul Barry_, Jun 26 2008
%C Equals row sums of triangle A153341. - _Gary W. Adamson_, Dec 24 2008
%H Colin Barker, <a href="/A026581/b026581.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,4).
%F G.f.: (1 + 2*x) / (1 - x - 4*x^2).
%F a(n) = a(n-1) + 4*a(n-2), n>1.
%F a(n) = 2*A006131(n-1) + A006131(n), n>0.
%F a(n) = (2^(-1-n)*((1-sqrt(17))^n*(-5+sqrt(17)) + (1+sqrt(17))^n*(5+sqrt(17))))/sqrt(17). - _Colin Barker_, Dec 22 2016
%t CoefficientList[Series[(1+2x)/(1-x-4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{1,4},{1,3},30] (* _Harvey P. Dale_, Aug 04 2015 *)
%o (PARI) Vec((1+2*x)/(1-x-4*x^2) + O(x^30)) \\ _Colin Barker_, Dec 22 2016
%o (Magma) I:=[1,3]; [n le 2 select I[n] else Self(n-1) +4*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Aug 03 2019
%o (Sage) ((1+2*x)/(1-x-4*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 03 2019
%o (GAP) a:=[1,3];; for n in [3..30] do a[n]:=a[n-1]+4*a[n-2]; od; a; # _G. C. Greubel_, Aug 03 2019
%Y Cf. A006131, A026568, A026583, A026597, A026599, A052923, A055099.
%Y Cf. A153341. - _Gary W. Adamson_, Dec 24 2008
%K nonn,easy
%O 0,2
%A _Clark Kimberling_
%E Edited by _Ralf Stephan_, Jul 20 2013
|
