cp's OEIS Frontend

A027003 a(n) = A026998(2*n, n+4).

%I A027003 #11 Jul 23 2025 01:02:04
%S A027003 1,64,901,6821,36425,155793,573382,1899933,5844446,17056486,47974934,
%T A027003 131553646,354615679,945220982,2501450971,6590435731,17316698039,
%U A027003 45428211431,119066290172,311909267867,816853717452,2138914514428,5600243896572,14662288678348,38387242941837
%N A027003 a(n) = A026998(2*n, n+4).
%H A027003 G. C. Greubel, <a href="/A027003/b027003.txt">Table of n, a(n) for n = 4..1000</a>
%H A027003 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (11,-53,148,-266,322,-266,148,-53,11,-1).
%F A027003 From _G. C. Greubel_, Jul 20 2025: (Start)
%F A027003 a(n) = Lucas(2*n+9) - f(n), where f(n) = (48*n^7 - 280*n^6 + 1596*n^5 - 910*n^4 + 10122*n^3 + 20405*n^2 + 46509*n + 47880)/630.
%F A027003 G.f.: x^4*(1 + 53*x + 250*x^2 + 154*x^3 - 59*x^4 - 15*x^5)/((1-x)^8*(1-3*x+x^2)).
%F A027003 E.g.f.: 4*exp(3*x/2)*(19*cosh(p*x) + 17*p*sinh(p*x)) - (1/630)*(47880 + 77490*x + 62685*x^2 + 33810*x^3 + 13650*x^4 + 4116*x^5 + 728*x^6 + 48*x^7)*exp(x), where 2*p = sqrt(5). (End)
%t A027003 f[n_]:= (48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n+47880)/630;
%t A027003 A027003[n_]:= LucasL[2*n+9] -f[n];
%t A027003 Table[A027003[n], {n,4,50}] (* _G. C. Greubel_, Jul 20 2025 *)
%o A027003 (Magma)
%o A027003 A027003:= func< n | Lucas(2*n+9) -(48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n +47880)/630 >;
%o A027003 [A027003(n): n in [4..45]]; // _G. C. Greubel_, Jul 20 2025
%o A027003 (SageMath)
%o A027003 def A027003(n): return lucas_number2(2*n+9,1,-1) -(48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n +47880)//630
%o A027003 print([A027003(n) for n in range(4,46)]) # _G. C. Greubel_, Jul 20 2025
%Y A027003 Cf. A000032, A026998.
%K A027003 nonn
%O A027003 4,2
%A A027003 _Clark Kimberling_
%E A027003 More terms from _Sean A. Irvine_, Oct 21 2019