A053088 a(n) = 3*a(n-2) + 2*a(n-3) for n > 2, a(0)=1, a(1)=0, a(2)=3.

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

Offset

0,3

Comments

Growth of happy bug population in GCSE math 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

From Paul Curtz, Nov 02 2021 (Start)

a(n-2) difference table (from 0, 0, a(n)):

0 0 1 0 3 2 9 12 31 54 ...

0 1 -1 3 -1 7 3 19 23 63 ...

1 -2 4 -4 8 -4 16 4 40 44 ...

-3 6 -8 12 -12 20 -12 36 4 84 ...

9 -14 20 -24 32 -32 48 -32 80 0 ...

-23 34 -44 56 -64 80 -80 112 -80 176 ...

57 -78 100 -120 144 -160 192 -192 256 -192 ...

... .

The signature is valid for every row.

a(n-2) + a(n-1) = A001045(n).

a(n-2) + a(n+1) = A062510(n) = 3*A001045(n).

a(n-2) + a(n+3) = see A144472(n+1).

Second subdiagonal: 1, 6, 20, 56, 144, 352, ... = A014480(n).

First subdiagonal: -A036895(n) = -2*A001787(n).

Main diagonal: A001787(n) = -first and -third upper diagonals.

Second, fourth and fifth upper diagonals: A001792(n), A045891(n+2) and A172160(n+1). (End)

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.

Iwan Duursma, Xiao Li, and Hsin-Po Wang, Multilinear Algebra for Distributed Storage, arXiv:2006.08911 [cs.IT], 2020.

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} binomial(3*n,n)*x^n / (1+x)^(3*n+3). - Paul D. Hanna, Mar 03 2012

E.g.f.: 1 + (1/9)*(exp(-x)*(3*x - 2) + 2*exp(2*x)). - Stefano Spezia, Sep 27 2019

Mathematica

CoefficientList[Series[1/(1 - 3 x^2 - 2 x^3), {x, 0, 32}], x] (* Michael De Vlieger, Sep 30 2019 *)

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.

Cf. A001045, A001787, A001792, A014480, A036895, A062510, A045891, A172160.

Sequence in context: A038220 A303901 A053151 * A077898 A303631 A358603

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