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

1,1

Links

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

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A027960, A360278.

Sequence in context: A132657 A379628 A161784 * A078799 A203288 A026957

Adjacent sequences: A027982 A027983 A027984 * A027986 A027987 A027988

Keyword

nonn

Author

Clark Kimberling

Status

approved