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A099232 a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Table of n, a(n) for n=0..30.

Index entries for linear recurrences with constant coefficients, signature (2,2,-3).

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

CROSSREFS

Cf. A001045, A006130.

Sequence in context: A263899 A062424 A258344 * A280758 A053562 A003039

Adjacent sequences:  A099229 A099230 A099231 * A099233 A099234 A099235

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Oct 08 2004

STATUS

approved

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Last modified July 4 17:52 EDT 2022. Contains 355083 sequences. (Running on oeis4.)