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

0,4

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.

Links

Table of n, a(n) for n=0..30.

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000045, A001654, A007598.

Sequence in context: A320203 A217093 A278639 * A222643 A129014 A335412

Adjacent sequences: A350784 A350785 A350786 * A350788 A350789 A350790

Keyword

nonn,easy

Author

Greg Dresden, Jan 16 2022

Status

approved