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

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

Offset

0,2

Comments

In general, for expansion of 1/((1-bx)(1-cx)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - bc*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).

Formula

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

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

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

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

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

Mathematica

Join[{a = 1, b = 11}, Table[c = 11b - 24a; a = b; b = c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
CoefficientList[Series[1 / ((1 - 3 x) (1 - 8 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 24 2013 *)
LinearRecurrence[{11, -24}, {1, 11}, 30] (* Harvey P. Dale, Feb 03 2022 *)

Prog

(Sage) [lucas_number1(n, 11, 24) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009
(PARI) Vec(1/((1-3*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
(Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Crossrefs

Cf. A190543.

Sequence in context: A137017 A133833 A142260 * A138930 A106136 A223622

Adjacent sequences: A016137 A016138 A016139 * A016141 A016142 A016143

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved