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

0,2

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,4).

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
(Sage) ((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

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

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

Sequence in context: A029855 A209397 A110014 * A151535 A181360 A001372

Adjacent sequences: A026578 A026579 A026580 * A026582 A026583 A026584

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Edited by Ralf Stephan, Jul 20 2013

Status

approved