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%I #22 Sep 08 2022 08:44:45
%S 1,27,493,7611,107293,1432011,18457741,232505307,2883927805,
%T 35398400235,431393410669,5231599117563,63232056214237,
%U 762504498009099,9180490786688077,110414131486397979,1326988747136473789
%N Expansion of 1/((1-7*x)*(1-8*x)*(1-12*x)).
%H Indranil Ghosh, <a href="/A020969/b020969.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (27,-236,672).
%F a(n) = 27*a(n-1) - 236*a(n-2) + 672*a(n-3), n>=3. - _Vincenzo Librandi_, Mar 15 2011
%F a(n) = 20*a(n-1) - 96*a(n-2) + 7^n for n>1, a(0)=1, a(1)=27. - _Vincenzo Librandi_, Mar 15 2011
%F a(n) = (7^(n+2) - 10*8^(n+1) + 3*12^(n+1))/5. - _Bruno Berselli_, Mar 15 2011
%e a(5) = (7^(5 + 2) - 10*8^(5 + 1) + 3*12^(5 + 1))/5 = (7^7 - 10*8^6 + 3*12 ^ 6)/5 = 7160055/5 = 1432011. - _Indranil Ghosh_, Feb 28 2017
%t CoefficientList[Series[1/((1 - 7 x) (1 - 8 x) (1 - 12 x)), {x, 0, 16}], x] (* or *) LinearRecurrence[{27, -236, 672}, {1, 27, 493}, 17] (* or *) Table[(7^(n + 2) - 10 8^(n + 1) + 3 12^(n + 1))/5, {n, 0, 16}] (* _Indranil Ghosh_, Feb 28 2017 *)
%o (PARI) a(n) = (7^(n+2)-10*8^(n+1)+3*12^(n+1))/5; \\ _Indranil Ghosh_, Feb 28 2017
%o (Python) def A020969(n): return (7**(n+2)-10*8**(n+1)+3*12**(n+1))/5 # _Indranil Ghosh_, Feb 28 2017
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-7*x)*(1-8*x)*(1-12*x)))); // _G. C. Greubel_, May 31 2018
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_