A027985 a(n) = Sum_{k=0..2*n-1} T(n, k)*T(n, k+1), T given by A027960.

6, 35, 144, 564, 2186, 8468, 32856, 127729, 497454, 1940525, 7580656, 29651385, 116111194, 455138499, 1785707924, 7011933544, 27554583254, 108355491404, 426368213364, 1678704356644, 6613026412314, 26064305550054, 102777232982624

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A027960, A360278.

Programs

Magma

f:= func< n,k | (&+[Binomial(2*n-k+j,j)*Lucas(2*(k-n-j)): j in [0..k-n-1]]) >;
A027960:= func< n,k | k le n select Lucas(k+1) else Lucas(k+1) - f(n,k) >;
A027985:= func< n | (&+[A027960(n,k)*A027960(n,k+1): k in [0..2*n-1]]) >;
[A027985(n): n in [1..40]]; // G. C. Greubel, Jun 13 2025

Mathematica

f[n_, k_]:= f[n,k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
A027960[n_, k_]:= LucasL[k+1] - f[n,k]*Boole[k>n];
A027985[n_]:= A027985[n]= Sum[A027960[n,k]*A027960[n,k+1], {k,0,2*n-1}];
Table[A027985[n], {n,40}] (* G. C. Greubel, Jun 13 2025 *)

SageMath

def L(n): return lucas_number2(n,1,-1)
def f(n,k): return sum(binomial(2*n-k+j,j)*L(2*(k-n-j)) for j in range(k-n))
def A027960(n,k): return L(k+1) - f(n,k)*int(k>n)
def A027985(n): return sum(A027960(n,k)*A027960(n,k+1) for k in range(2*n))
print([A027985(n) for n in range(1,41)]) # G. C. Greubel, Jun 13 2025

Formula

a(n) = A360278(n) - 1 + Sum_{k=n..2*n-1} A027960(n,k)*A027960(n,k+1), for n >= 1. - G. C. Greubel, Jun 13 2025