%I #55 Sep 22 2025 16:00:24

%S 1,11,97,803,6505,52283,418993,3354131,26839609,214736555,1717951489,

%T 13743789059,109950843913,879608345627,7036871547985,56294986732787,

%U 450359936909017,3602879624412299,28823037382718881,230584300224012515,1844674405278884521,14757395252691429371,118059162052912494577,944473296517443135443

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

%C 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

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

%H Vincenzo Librandi, <a href="/A016140/b016140.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-24).

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

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

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

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

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

%F E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - _G. C. Greubel_, Nov 14 2024

%t Table[(8^(n+1)-3^(n+1))/5, {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2011 *)

%t CoefficientList[Series[1/((1-3 x)(1-8 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jun 24 2013 *)

%t LinearRecurrence[{11,-24},{1,11},30] (* _Harvey P. Dale_, Feb 03 2022 *)

%o (SageMath) [lucas_number1(n,11,24) for n in range(1, 30)] # _Zerinvary Lajos_, Apr 27 2009

%o (PARI) Vec(1/((1-3*x)*(1-8*x))+O(x^30)) \\ _Charles R Greathouse IV_, Sep 23 2012

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // _Vincenzo Librandi_, Jun 24 2013

%Y 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).

%Y Cf. A190543.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_