This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A026581 #37 Jun 02 2025 20:48:33
%S A026581 1,3,7,19,47,123,311,803,2047,5259,13447,34483,88271,226203,579287,
%T A026581 1484099,3801247,9737643,24942631,63893203,163663727,419236539,
%U A026581 1073891447,2750837603,7046403391,18049753803,46235367367,118434382579,303375852047,777113382363
%N A026581 Expansion of (1 + 2*x) / (1 - x - 4*x^2).
%C A026581 T(n,0) + T(n,1) + ... + T(n,2n), T given by A026568.
%C A026581 Row sums of Riordan array ((1+2x)/(1+x),x(1+2x)/(1+x)). Binomial transform is A055099. - _Paul Barry_, Jun 26 2008
%C A026581 Equals row sums of triangle A153341. - _Gary W. Adamson_, Dec 24 2008
%C A026581 Also, the number of walks of length n starting at vertex 0 in the graph with 4 vertices and edges {{0,1}, {0,2}, {0,3}, {1,2}, {2,3}}. - _Sean A. Irvine_, Jun 02 2025
%H A026581 Colin Barker, <a href="/A026581/b026581.txt">Table of n, a(n) for n = 0..1000</a>
%H A026581 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,4).
%F A026581 G.f.: (1 + 2*x) / (1 - x - 4*x^2).
%F A026581 a(n) = a(n-1) + 4*a(n-2), n>1.
%F A026581 a(n) = 2*A006131(n-1) + A006131(n), n>0.
%F A026581 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 A026581 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 A026581 (PARI) Vec((1+2*x)/(1-x-4*x^2) + O(x^30)) \\ _Colin Barker_, Dec 22 2016
%o A026581 (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 A026581 (Sage) ((1+2*x)/(1-x-4*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 03 2019
%o A026581 (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 A026581 Cf. A006131, A026568, A026583, A026597, A026599, A052923, A055099.
%Y A026581 Cf. A153341. - _Gary W. Adamson_, Dec 24 2008
%K A026581 nonn,easy
%O A026581 0,2
%A A026581 _Clark Kimberling_
%E A026581 Edited by _Ralf Stephan_, Jul 20 2013