cp's OEIS Frontend

A027006 a(n) = T(2*n+1, n+3), T given by A026998.

%I A027006 #13 Jul 23 2025 00:57:24
%S A027006 1,34,281,1397,5353,17643,53062,150833,414210,1114160,2960806,7814074,
%T A027006 20544191,53902532,141273663,370060623,969088727,2537431693,
%U A027006 6643486220,17393369595,45537037936,119218243314,312118286876,817137321092,2139294503373,5600747154678
%N A027006 a(n) = T(2*n+1, n+3), T given by A026998.
%C A027006 Bisection of A053298.
%H A027006 Colin Barker, <a href="/A027006/b027006.txt">Table of n, a(n) for n = 2..1000</a>
%H A027006 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (8,-26,45,-45,26,-8,1).
%F A027006 G.f.: x^2*(1+26*x+35*x^2-12*x^3-2*x^4) / ((1-x)^5*(1-3*x+x^2)). - _Colin Barker_, Feb 19 2016
%F A027006 From _G. C. Greubel_, Jul 22 2025: (Start)
%F A027006 a(n) = Lucas(2*n+7) - (12*n^4 + 20*n^3 + 81*n^2 + 169*n + 174)/6.
%F A027006 E.g.f.: exp(3*x/2)*(29*cosh(p*x) + 26*p*sinh(p*x)) - (1/6)*(174 + 282*x + 225*x^2 + 92*x^3 + 12*x^4)*exp(x), where 2*p = sqrt(5). (End)
%t A027006 A027006[n_]:= LucasL[2*n+7] -(12*n^4 +20*n^3 +81*n^2 +169*n +174)/6;
%t A027006 Table[A027006[n], {n,2,42}] (* _G. C. Greubel_, Jul 22 2025 *)
%o A027006 (PARI) Vec(x^2*(1+26*x+35*x^2-12*x^3-2*x^4) / ((1-x)^5*(1-3*x+x^2)) + O(x^40)) \\ _Colin Barker_, Feb 19 2016
%o A027006 (Magma)
%o A027006 A027006:= func< n | Lucas(2*n+7) -(12*n^4 +20*n^3 +81*n^2 +169*n +174)/6 >;
%o A027006 [A027006(n): n in [2..40]]; // _G. C. Greubel_, Jul 22 2025
%o A027006 (SageMath)
%o A027006 def A027006(n): return lucas_number2(2*n+7,1,-1) -(12*n^4 +20*n^3 +81*n^2 +169*n +174)//6
%o A027006 print([A027006(n) for n in range(2,41)]) # _G. C. Greubel_, Jul 22 2025
%Y A027006 Cf. A000032, A026998, A053298.
%K A027006 nonn,easy
%O A027006 2,2
%A A027006 _Clark Kimberling_