A019943 - OEIS

%I #20 Jan 23 2025 10:24:44

%S 1,21,298,3570,38971,401751,3988468,38583300,366449941,3434404281,

%T 31873887838,293663563830,2690806228111,24553315831611,

%U 223338364450408,2026585451393160,18355202849805481,166009125098571741,1499772036736668178,13537796780062999290

%N Expansion of 1/((1-5*x)*(1-7*x)*(1-9*x)).

%H Vincenzo Librandi, <a href="/A019943/b019943.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 (21,-143,315).

%F a(n) = (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8. - _R. J. Mathar_, Jun 29 2013

%F a(0)=1, a(1)=21, a(2)=298; for n>2, a(n) = 21*a(n-1) -143*a(n-2) +315*a(n-3). - _Vincenzo Librandi_, Jul 03 2013

%F a(n) = 16*a(n-1) -63*a(n-2) +5^n. - _Vincenzo Librandi_, Jul 03 2013

%F E.g.f.: (25*exp(5*x) - 98*exp(7*x) + 81*exp(9*x))/8. - _G. C. Greubel_, Nov 24 2018

%p a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [21, -143, 315][i], 0)))^n)[1, 1]: seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 03 2013

%t CoefficientList[Series[1/((1-5x)(1-7x)(1-9 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 03 2013 *)

%t LinearRecurrence[{21,-143,315}, {1, 21, 298}, 20] (* or *) Table[(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8, {n,0,20}] (* _G. C. Greubel_, Nov 24 2018 *)

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-5*x)*(1-7*x)*(1-9*x)))); // _Vincenzo Librandi_, Jul 03 2013

%o (Magma) I:=[1, 21, 298]; [n le 3 select I[n] else 21*Self(n-1)-143*Self(n-2)+315*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 03 2013

%o (Magma) [(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8: n in [0..20]]; // _G. C. Greubel_, Nov 24 2018

%o (PARI) vector(20, n, n--; (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8) \\ _G. C. Greubel_, Nov 24 2018

%o (Sage) [(5^(n+2) - 2*7^(n+2) + 9^(n+2))/8 for n in range(20)] # _G. C. Greubel_, Nov 24 2018

%o (GAP) List([0..20], n -> (5^(n+2) - 2*7^(n+2) + 9^(n+2))/8); # _G. C. Greubel_, Nov 24 2018

%K nonn

%O 0,2

%A _N. J. A. Sloane_