A027984 a(n) = Sum_{k=0..n} T(n, k)*T(n, n+k), T given by A027960.

1, 6, 20, 58, 161, 436, 1165, 3088, 8146, 21426, 56255, 147538, 386681, 1013026, 2653240, 6948058, 18193141, 47634936, 124717445, 326526748, 854877926, 2238131506, 5859556195, 15340601158, 40162350961, 105146619486, 275277778940

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Crossrefs

Cf. A000032, A000045, A027960.

Programs

Magma

A027984:= func< n | (5*Fibonacci(2*n+2) +Lucas(2*n+1) -Lucas(n+3))/2 >;
[A027984(n): n in [0..40]]; // G. C. Greubel, Jun 10 2025

Mathematica

LinearRecurrence[{4,-3,-2,1}, {1,6,20,58}, 41] (* G. C. Greubel, Jun 10 2025 *)

PARI

T027960(r,n) = if(r<0||n>2*r, return(0)); if(n==0||n==2*r, return(1)); if(n==1, 3, T027960(r-1, n-1)+T027960(r-1, n-2));
a(n) = sum(k=0, n, T027960(n, k)*T027960(n, n+k)); \\ Michel Marcus, Feb 25 2015

SageMath

L = lucas_number2
def A027984(n): return (5*fibonacci(2*n+2) +L(2*n+1,1,-1) -L(n+3,1,-1))//2
print([A027984(n) for n in range(41)]) # G. C. Greubel, Jun 10 2025

Formula

Conjectures from Colin Barker, Feb 25 2015: (Start)

a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4).

G.f.: (1-x)*(1+x)*(1+2*x) / ((1-3*x+x^2)*(1-x-x^2)). (End)

a(n) = (1/2)*(5*A000045(2*n+2) + A000032(2*n+1) - A000032(n+3)). - G. C. Greubel, Jun 10 2025