A016140 Expansion of 1/((1-3*x)*(1-8*x)).

1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, 1717951489, 13743789059, 109950843913, 879608345627, 7036871547985, 56294986732787, 450359936909017, 3602879624412299, 28823037382718881, 230584300224012515, 1844674405278884521, 14757395252691429371, 118059162052912494577, 944473296517443135443

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

In general, for expansion of 1/((1-b*x)*(1-c*x)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - b*c*a(n-2) = b*a(n-1) + c^n = c*a(n-1) + b^n = Sum_{i=0..n} b^i*c^(n-i). - Henry Bottomley, Jul 20 2000

8*a(n) gives the number of edges in the n-th-order Sierpiński carpet graph. - Eric W. Weisstein, Aug 19 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (11,-24).

Crossrefs

Sequences with g.f. 1/((1-n*x)*(1-8*x)): A001018 (n=0), A023001 (n=1), A016131 (n=2), this sequence (n=3), A016152 (n=4), A016162 (n=5), A016170 (n=6), A016177 (n=7), A053539 (n=8), A016185 (n=9), A016186 (n=10), A016187 (n=11), A016188 (n=12), A060195 (n=16).

Cf. A190543.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Mathematica

Table[(8^(n+1)-3^(n+1))/5, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
CoefficientList[Series[1/((1-3 x)(1-8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 24 2013 *)
LinearRecurrence[{11,-24},{1,11},30] (* Harvey P. Dale, Feb 03 2022 *)

PARI

Vec(1/((1-3*x)*(1-8*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

Sage

[lucas_number1(n,11,24) for n in range(1, 30)] # Zerinvary Lajos, Apr 27 2009

Formula

a(n) = (8^(n+1) - 3^(n+1))/5.

a(n) = 11*a(n-1) - 24*a(n-2).

a(n) = 3*a(n-1) + 8^n.

a(n) = 8*a(n-1) + 3^n.

a(n) = Sum_{i=0..n} 3^i*8^(n-i).

E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - G. C. Greubel, Nov 14 2024