A085287 Expansion of g.f. (1+4*x)/((1-x)*(1+x)*(1-3*x)).

1, 7, 22, 70, 211, 637, 1912, 5740, 17221, 51667, 155002, 465010, 1395031, 4185097, 12555292, 37665880, 112997641, 338992927, 1016978782, 3050936350, 9152809051, 27458427157, 82375281472, 247125844420, 741377533261, 2224132599787, 6672397799362, 20017193398090

Offset: 0

Author

Paul Barry, Jun 26 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A084431.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Crossrefs

Cf. A084431.

Programs

Magma

[(-10-3*(-1)^n+21*3^n)/8: n in [0..30]]; // Vincenzo Librandi, Nov 16 2011

Maple

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+5 od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 14 2008

Mathematica

LinearRecurrence[{3,1,-3},{1,7,22},30] (* Harvey P. Dale, Sep 22 2023 *)

PARI

Vec((1+4*x)/((1-x^2)*(1-3*x)) + O(x^30)) \\ Michel Marcus, Aug 14 2017

Formula

a(n) = (-10 - 3(-1)^n + 21*3^n)/8.

a(n) = 2*a(n-1) + 3*a(n-2) + 5, a(0)=0, a(1)=1. - Zerinvary Lajos, Dec 14 2008

From Stefano Spezia, Sep 20 2023: (Start)

a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n > 2.

E.g.f.: exp(x)*(9*cosh(2*x) + 12*sinh(2*x) - 5)/4. (End)