A016208 Expansion of 1/((1-x)*(1-3*x)*(1-4*x)).

1, 8, 45, 220, 1001, 4368, 18565, 77540, 320001, 1309528, 5326685, 21572460, 87087001, 350739488, 1410132405, 5662052980, 22712782001, 91044838248, 364760483725, 1460785327100, 5848371485001, 23409176469808, 93683777468645, 374876324642820, 1499928942876001

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Binomial transform of A085277. - Paul Barry, Jun 25 2003

Number of walks of length 2n+5 between two nodes at distance 5 in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..1000
• Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, Brauer Configuration Algebras and Their Applications in Graph Energy Theory, Mathematics (2021) Vol. 9, 3042.
• Index entries for linear recurrences with constant coefficients, signature (8,-19,12)

Crossrefs

Cf. A000225, A000295, A000392, A002275, A003462, A003463, A003464, A023000, A023001, A002452, A016123, A016125, A016256.

Programs

GAP

a:=[1,8,45];; for n in [4..30] do a[n]:=8*a[n-1]-19*a[n-2]+12*a[n-3]; od; Print(a); # Muniru A Asiru, Apr 19 2019

Mathematica

Table[(2^(2*n + 3) - 3^(n + 2) + 1)/6, {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-x)(1-3x)(1-4x)),{x,0,30}],x] (* or *) LinearRecurrence[ {8,-19,12},{1,8,45},30] (* Harvey P. Dale, Apr 09 2012 *)

PARI

Vec(1/((1-x)*(1-3*x)*(1-4*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Formula

a(n) = 16*4^n/3 + 1/6 - 9*3^n/2. - Paul Barry, Jun 25 2003

a(0) = 0, a(1) = 8, a(n) = 7*a(n-1) - 12*a(n-2) + 1. - Vincenzo Librandi, Feb 10 2011

a(0) = 1, a(1) = 8, a(2) = 45, a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3). - Harvey P. Dale, Apr 09 2012