cp's OEIS Frontend

A027018 a(n) = T(2*n+1, n+3), T given by A027011.

%I A027018 #12 Jun 16 2025 08:48:51
%S A027018 1,9,150,1085,5283,20495,69007,212020,613633,1708508,4640978,12414802,
%T A027018 32903418,86731043,227905816,597838223,1566763325,4103989113,
%U A027018 10747219441,28140274566,73676929931,192894712070,505012447636,1322149114676,3461442847524,9062189100301
%N A027018 a(n) = T(2*n+1, n+3), T given by A027011.
%H A027018 Colin Barker, <a href="/A027018/b027018.txt">Table of n, a(n) for n = 2..1000</a>
%H A027018 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (9,-34,71,-90,71,-34,9,-1).
%F A027018 G.f.: x^2*(1+103*x^2-30*x^3+69*x^4-73*x^5+34*x^6-9*x^7+x^8) / ((1-x)^6*(1-3*x+x^2)). - _Colin Barker_, Feb 19 2016
%F A027018 From _G. C. Greubel_, Jun 16 2025: (Start)
%F A027018 a(n) = A000032(2*n+8) - (1/30)*(1410 + 1351*n + 655*n^2 + 230*n^3 + 20*n^4 + 24*n^5) + [n=2].
%F A027018 E.g.f.: exp(3*x/2)*( 47*cosh(sqrt(5)*x/2) + 21*sqrt(5)*sinh(sqrt(5)*x/2) ) + x^2/2 - (1/30)*(1410 + 2280*x + 1845*x^2 + 950*x^3 + 260*x^4 + 24*x^5)*exp(x). (End)
%t A027018 Table[LucasL[2*n+8] -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)/30 + Boole[n==2], {n,2,50}] (* _G. C. Greubel_, Jun 16 2025 *)
%o A027018 (PARI) Vec(x^2*(1+103*x^2-30*x^3+69*x^4-73*x^5+34*x^6-9*x^7+x^8)/((1-x)^6*(1-3*x+x^2)) + O(x^40)) \\ _Colin Barker_, Feb 19 2016
%o A027018 (Magma)
%o A027018 A027018:= func< n | n eq 2 select 1 else Lucas(2*n+8) -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)/30 >;
%o A027018 [A027018(n): n in [2..50]]; // _G. C. Greubel_, Jun 16 2025
%o A027018 (SageMath)
%o A027018 def A027018(n): return lucas_number2(2*n+8,1,-1) -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)//30 + int(n==2)
%o A027018 print([A027018(n) for n in range(2,51)]) # _G. C. Greubel_, Jun 16 2025
%Y A027018 Cf. A000032, A027011.
%K A027018 nonn,easy
%O A027018 2,2
%A A027018 _Clark Kimberling_