A350787 Convolution of A001654 and A007598.

0, 0, 1, 3, 12, 38, 122, 372, 1119, 3301, 9624, 27756, 79380, 225384, 636061, 1785639, 4990116, 13889618, 38524238, 106514652, 293668923, 807608137, 2215854384, 6066935640, 16579195560, 45226399440, 123173004985, 334955873739, 909611388732, 2466965351678, 6682629071522

Offset: 0

Author

Greg Dresden, Jan 16 2022

Keywords

• nonn
• easy

Comments

Note that A001654(n) = F(n)*F(n+1) and A007598(n) = F(n)^2, for F(n) = A000045(n), the n-th Fibonacci number.

Examples

			For n=2, a(2) = F(0)*F(1)*F(2)^2 + F(1)*F(2)*F(1)^2 + F(2)*F(3)*F(0)^2 = 1.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,0,-10,0,4,-1).

Crossrefs

Cf. A000045, A001654, A007598.

Programs

Mathematica

Table[Sum[Fibonacci[i]*Fibonacci[i + 1]*Fibonacci[n - i]^2, {i, 0, n}], {n, 0, 30}]

PARI

a(n) = sum(i=0, n, fibonacci(i)*fibonacci(i+1)*fibonacci(n-i)^2); \\ Michel Marcus, Jan 17 2022

Formula

a(n) = Sum_{i=0..n} F(i)*F(i+1)*F(n-i)^2.

a(n) = ((n + 2)/5)*F(n)*F(n+1) - (3/25)*(F(2*n+2) + (n + 1)*(-1)^(n + 1)).

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

a(n) = 4*a(n-1) - 10*a(n-3) + 4*a(n-5) - a(n-6) for n > 5. - Amiram Eldar, Jan 17 2022