A099232 a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3), with a(0) = 0, a(1) = 1, a(2) = 2.

0, 1, 2, 6, 13, 32, 72, 169, 386, 894, 2053, 4736, 10896, 25105, 57794, 133110, 306493, 705824, 1625304, 3742777, 8618690, 19847022, 45703093, 105244160, 242353440, 558085921, 1285146242, 2959404006, 6814842733, 15693054752, 36137582952

Offset: 0

Author

Paul Barry, Oct 08 2004

Keywords

• easy
• nonn

Comments

Partial sums of A006130 (with leading zero).

Specify a triangle by T(n,0) = T(n+1,1) = A001045(n) and T(n,k) = T(n-1,k-1) + T(n-1,k-2) + T(n-2,k-2) otherwise. Then T(n,n)= a(n-1). - J. M. Bergot, May 24 2013

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-3).

Crossrefs

Cf. A001045, A006130.

Programs

Mathematica

LinearRecurrence[{2,2,-3},{0,1,2},40] (* Harvey P. Dale, Mar 02 2024 *)

Formula

G.f.: x/((1-x)*(1-x-3*x^2)).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k+1)*3^k.

a(n) = (1/2 + sqrt(13)/2)^n*(1/6 + 7*sqrt(13)/78) + (1/6 - 7*sqrt(13)/78)*(1/2 - sqrt(13)/2)^n - 1/3.

a(n+1) = Sum_{k=0..n} C(k+1,n-k+1)*3^(n-k). - Paul Barry, May 21 2006

a(n) = a(n-1) + 3*a(n-2) + 1, n > 1. - Gary Detlefs, Jun 21 2010

G.f.: Q(0)*x/(2-2*x), where Q(k) = 1 + 1/(1 - x*(4*k+1 + 3*x)/( x*(4*k+3 + 3*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013