A027986 a(n) = Sum_{k=0..2*n-2} T(n, k)*T(n, k+2), T given by A027960.

20, 100, 420, 1694, 6746, 26735, 105722, 417613, 1648692, 6507197, 25681362, 101359219, 400094756, 1579568360, 6237401648, 24635828774, 97327071806, 384596629610, 1520138671688, 6009885957464, 23765835521966, 94003008751940

Offset

2,1

Links

G. C. Greubel, Table of n, a(n) for n = 2..1000

Formula

a(n) = A000032(2*n+1) - 7 + 3*(n mod 2) + Sum_{k=n-1..2*n-2} A027960(n, k)*A027960(n, k+2). - G. C. Greubel, Jun 14 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];
A027986[n_]:= A027986[n]= Sum[A027960[n, k]*A027960[n, k+2], {k, 0, 2*n-2}];
Table[A027986[n], {n, 2, 40}] (* G. C. Greubel, Jun 14 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) >;
A027986:= func< n | (&+[A027960(n, k)*A027960(n, k+2): k in [0..2*n-2]]) >;
[A027986(n): n in [2..40]]; // G. C. Greubel, Jun 14 2025
(SageMath)
@CachedFunction
def T(n, k): # T = A027960
    if (k>2*n): return 0
    elif (k<n+1): return lucas_number2(k+1, 1, -1)
    else: return T(n-1, k-2) + T(n-1, k-1)
def A027986(n): return sum(T(n, k)*T(n, k+2) for k in range(2*n-1))
print([A027986(n) for n in range(2, 41)]) # G. C. Greubel, Jun 14 2025

Crossrefs

Cf. A027960.

Sequence in context: A039455 A294112 A188050 * A240437 A174078 A396458

Adjacent sequences: A027983 A027984 A027985 * A027987 A027988 A027989

Keyword

nonn

Author

Clark Kimberling

Status

approved