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%I #20 Sep 08 2022 08:44:44
%S 1,23,367,5075,65551,817643,10013527,121451315,1465540351,17637184763,
%T 211960186087,2545454874755,30557298487951,366759842503883,
%U 4401557777453047,52821361851453395,633872505937432351
%N Expansion of 1/((1-5*x)*(1-6*x)*(1-12*x)).
%C a(n) is the number of partitions of n into parts 5, 6, and 12. - _Joerg Arndt_, Apr 29 2017
%H Vincenzo Librandi, <a href="/A019869/b019869.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (23,-162,360).
%F G.f.: 1/((1-5*x)*(1-6*x)*(1-12*x)).
%F a(n) = 25*5^n/7-6*6^n+24*12^n/7. - _R. J. Mathar_, Jun 29 2013
%F a(0)=1, a(1)=23, a(2)=367; for n>2, a(n) = 23*a(n-1) -162*a(n-2) +360*a(n-3). - _Vincenzo Librandi_, Jul 03 2013
%F a(n) = 18*a(n-1) -72*a(n-2) +5^n. - _Vincenzo Librandi_, Jul 03 2013
%p A019869:=n->25*5^n/7-6*6^n+24*12^n/7: seq(A019869(n), n=0..25); # _Wesley Ivan Hurt_, Apr 28 2017
%t CoefficientList[Series[1 / ((1 - 5 x) (1 - 6 x) (1 - 12 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 03 2013 *)
%t LinearRecurrence[{23,-162,360},{1,23,367},20] (* _Harvey P. Dale_, Aug 04 2020 *)
%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-5*x)*(1-6*x)*(1-12*x)))); /* or */ I:=[1, 23, 367]; [n le 3 select I[n] else 23*Self(n-1)-162*Self(n-2)+360*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 03 2013
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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