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A020579 Expansion of g.f. 1/((1-6*x)*(1-8*x)*(1-9*x)).

%I A020579 #21 Mar 26 2025 18:28:53
%S A020579 1,23,355,4595,53851,592403,6240235,63710915,635468251,6225852083,
%T A020579 60146237515,574587484835,5439634923451,51116555484563,
%U A020579 477406092913195,4435981769620355,41041272503703451,378327871809737843,3476703760455563275,31864966517183461475,291385416197758352251
%N A020579 Expansion of g.f. 1/((1-6*x)*(1-8*x)*(1-9*x)).
%H A020579 Vincenzo Librandi, <a href="/A020579/b020579.txt">Table of n, a(n) for n = 0..200</a>
%H A020579 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (23,-174,432).
%F A020579 a(n) = 6*6^n - 32*8^n + 27*9^n. - _R. J. Mathar_, Jun 30 2013
%F A020579 From _Vincenzo Librandi_, Jul 04 2013: (Start)
%F A020579 a(0)=1, a(1)=23, a(2)=355; for n>2, a(n) = 23*a(n-1) - 174*a(n-2) + 432*a(n-3).
%F A020579 a(n) = 17*a(n-1) - 72*a(n-2) + 6^n. (End)
%F A020579 From _Elmo R. Oliveira_, Mar 26 2025: (Start)
%F A020579 E.g.f.: exp(6*x)*(6 - 32*exp(2*x) + 27*exp(3*x)).
%F A020579 a(n) = A016172(n+1) - A016170(n+1). (End)
%t A020579 CoefficientList[Series[1 / ((1 - 6 x) (1 - 8 x) (1 - 9 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 04 2013 *)
%o A020579 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-6*x)*(1-8*x)*(1-9*x)))); // _Vincenzo Librandi_, Jul 04 2013
%o A020579 (Magma) I:=[1,23,355]; [n le 3 select I[n] else 23*Self(n-1)-174*Self(n-2)+432*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 04 2013
%Y A020579 Cf. A016170, A016172.
%K A020579 nonn,easy
%O A020579 0,2
%A A020579 _N. J. A. Sloane_