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

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A232603, A232604.

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

Programs

Mathematica

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

PARI

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

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

Extensions

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