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A053088 a(n) = 3*a(n-2) + 2*a(n-3) for n>2, a(0)=1, a(1)=0, a(2)=3. 11
1, 0, 3, 2, 9, 12, 31, 54, 117, 224, 459, 906, 1825, 3636, 7287, 14558, 29133, 58248, 116515, 233010, 466041, 932060, 1864143, 3728262, 7456549, 14913072, 29826171, 59652314, 119304657, 238609284, 477218599, 954437166, 1908874365 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Growth of happy bug population in GCSE maths course work assignment.

The generalized (3,2)-Padovan sequence p(3,2;n). See the W. Lang link under A000931. [Wolfdieter Lang, Jun 25 2010]

With offset 1: a(n) = -2^n*sum( k=0..n, k^p*q^k ) for p=1, q=-1/2. See also A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2). - Stanislav Sykora, Nov 27 2013

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

Index entries for linear recurrences with constant coefficients, signature (0, 3, 2).

FORMULA

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

With offset 1: a(1)=1; a(n) = 2*a(n-1)-(-1)^n*n; a(n) = (1/9)*(2^(n+1)-(-1)^n*(3*n+2)). - Benoit Cloitre, Nov 02 2002

a(n) = sum( k=0..floor(n/2), A078008(n-2k) ). - Paul Barry, Nov 24 2003

a(n) = sum( k=0..floor(n/2), binomial(k, n-2k)3^k*(2/3)^(n-2k) ). - Paul Barry, Oct 16 2004

a(n) = sum( k=0..n, A078008(k)*(1-(-1)^(n+k-1))/2 ). - Paul Barry, Apr 16 2005

a(n) = ( 2^(n+2) + (-1)^n*(3*n+5) )/9 (see also the B. Cloitre comment above). From the o.g.f. 1/(1-3*x^2-2*x^3) = 1/((1-2*x)*(1+x)^2) = (3/(1+x)^2 + 2/(1+x) + 4/(1-2*x))/9. [Wolfdieter Lang, Jun 25 2010]

From Wolfdieter Lang, Aug 26 2010: (Start)

a(n) = a(n-1) + 2*a(n-2) + (-1)^n for n>1, a(0)=1, a(1)=0.

Due to the identity for the o.g.f. A(x): A(x) = x*(1+2*x)*A(x) + 1/(1+x).

(This recurrence was observed by Gary Detlefs in a 08/25/10 e-mail to the author.) (End)

G.f.: sum( n=0..infinity, binomial(3*n,n)*x^n / (1+x)^(3*n+3) ). [Paul D. Hanna, Mar 03 2012]

PROG

(PARI) c(n)=(2^(n+1)-(-1)^n*(3*n+2))/9; a(n)=c(n+1); \\ Stanislav Sykora, Nov 27 2013

CROSSREFS

Cf. A232603, A232604.

Sequence in context: A038220 A303901 A053151 * A077898 A303631 A076584

Adjacent sequences:  A053085 A053086 A053087 * A053089 A053090 A053091

KEYWORD

nonn,easy

AUTHOR

Pauline Gorman (pauline(AT)gorman65.freeserve.co.uk), Feb 26 2000

EXTENSIONS

More terms from James A. Sellers, Feb 28 2000 and Christian G. Bower, Feb 29 2000.

STATUS

approved

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Last modified April 19 04:19 EDT 2019. Contains 322237 sequences. (Running on oeis4.)