A026581 Expansion of (1 + 2*x) / (1 - x - 4*x^2).

1, 3, 7, 19, 47, 123, 311, 803, 2047, 5259, 13447, 34483, 88271, 226203, 579287, 1484099, 3801247, 9737643, 24942631, 63893203, 163663727, 419236539, 1073891447, 2750837603, 7046403391, 18049753803, 46235367367, 118434382579, 303375852047, 777113382363

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

T(n,0) + T(n,1) + ... + T(n,2n), T given by A026568.

Row sums of Riordan array ((1+2x)/(1+x),x(1+2x)/(1+x)). Binomial transform is A055099. - Paul Barry, Jun 26 2008

Equals row sums of triangle A153341. - Gary W. Adamson, Dec 24 2008

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,4).

Crossrefs

Cf. A006131, A026568, A026583, A026597, A026599, A052923, A055099.

Cf. A153341. - Gary W. Adamson, Dec 24 2008

Programs

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

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

Mathematica

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 *)

PARI

Vec((1+2*x)/(1-x-4*x^2) + O(x^30)) \\ Colin Barker, Dec 22 2016

Sage

((1+2*x)/(1-x-4*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

Formula

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

a(n) = a(n-1) + 4*a(n-2), n>1.

a(n) = 2*A006131(n-1) + A006131(n), n>0.

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

Extensions

Edited by Ralf Stephan, Jul 20 2013