cp's OEIS Frontend

A027987 a(n) = Sum_{k=0..2*n-3} T(n, k)*T(n, k+3), T given by A027960.

%I A027987 #9 Jun 19 2025 19:07:22
%S A027987 58,272,1154,4757,19378,78422,315994,1269270,5086294,20345092,
%T A027987 81265106,324240331,1292556986,5149091916,20500830986,81586994576,
%U A027987 324577199086,1290904257426,5133037194706,20407032157936,81119882902118
%N A027987 a(n) = Sum_{k=0..2*n-3} T(n, k)*T(n, k+3), T given by A027960.
%H A027987 G. C. Greubel, <a href="/A027987/b027987.txt">Table of n, a(n) for n = 3..1000</a>
%F A027987 a(n) = A000032(n)^2 - 9 + Sum_{k=n-2..2*n-3} A027960(n,k)*A027960(n,k+3). - _G. C. Greubel_, Jun 14 2025
%t A027987 T[n_, k_]:= T[n,k]= If[k<n+1, LucasL[k+1], If[k>2*n, 0, T[n-1,k-1] + T[n-1,k-2]]];
%t A027987 A027987[n_]:= A027987[n]= Sum[T[n,k]*T[n,k+3], {k,0,2*n-3}]; (* T = A027960 *)
%t A027987 Table[A027987[n], {n,3,50}] (* _G. C. Greubel_, Jun 14 2025 *)
%o A027987 (Magma)
%o A027987 f:= func< n,k | (&+[Binomial(2*n-k+j,j)*Lucas(2*(k-n-j)): j in [0..k-n-1]]) >;
%o A027987 A027960:= func< n,k | k le n select Lucas(k+1) else Lucas(k+1) - f(n,k) >;
%o A027987 A027987:= func< n | (&+[A027960(n,k)*A027960(n,k+3): k in [0..2*n-3]]) >;
%o A027987 [A027987(n): n in [3..40]]; // _G. C. Greubel_, Jun 14 2025
%o A027987 (SageMath)
%o A027987 @CachedFunction
%o A027987 def T(n, k): # T = A027960
%o A027987     if (k>2*n): return 0
%o A027987     elif (k<n+1): return lucas_number2(k+1,1,-1)
%o A027987     else: return T(n-1, k-2) + T(n-1, k-1)
%o A027987 def A027987(n): return sum(T(n,k)*T(n,k+3) for k in range(2*n-2))
%o A027987 print([A027987(n) for n in range(2,41)]) # _G. C. Greubel_, Jun 14 2025
%Y A027987 Cf. A027960.
%K A027987 nonn
%O A027987 3,1
%A A027987 _Clark Kimberling_