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URL: https://oeis.org/A026581

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A026581
Expansion of (1 + 2*x) / (1 - x - 4*x^2).
11
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
0,2
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
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
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 *)
PROG
(PARI) Vec((1+2*x)/(1-x-4*x^2) + O(x^30)) \\ Colin Barker, Dec 22 2016
(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
(SageMath) ((1+2*x)/(1-x-4*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(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
CROSSREFS
KEYWORD
nonn,easy
EXTENSIONS
Edited by Ralf Stephan, Jul 20 2013
STATUS
approved