A354044 a(n) = 2*(-i)^n*(n*sin(c*(n+1)) - i*sin(-c*n))/sqrt(5) where c = arccos(i/2).

0, 2, 5, 11, 23, 45, 86, 160, 293, 529, 945, 1673, 2940, 5134, 8917, 15415, 26539, 45525, 77842, 132716, 225685, 382877, 648165, 1095121, 1846968, 3109850, 5228261, 8777315, 14716223, 24643389, 41220110, 68873848, 114964805, 191719849, 319436697, 531789785

Offset: 0

Author

Peter Luschny, May 16 2022

Keywords

• nonn
• easy

Links

• Jianing Song, Table of n, a(n) for n = 0..1000
• Peter Luschny, Illustration of A354044.
• Peter Luschny, The Fibonacci Function.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

Cf. A000045 (the Fibonacci numbers), A007502, A088209, A094588, A136391, A178521, A264147, A353595.

Programs

Julia

function fibrec(n::Int)
    n == 0 && return (BigInt(0), BigInt(1))
    a, b = fibrec(div(n, 2))
    c = a * (b * 2 - a)
    d = a * a + b * b
    iseven(n) ? (c, d) : (d, c + d)
end
function A354044(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n + 1)
    a*(n - 1) + b
end
println([A354044(n) for n in 0:35])

Maple

c := arccos(I/2): a := n -> 2*(-I)^n*(n*sin(c*(n+1)) - I*sin(-c*n))/sqrt(5):
seq(simplify(a(n)), n = 0..35);

PARI

a(n) = fibonacci(n) + n*fibonacci(n+1) \\ Jianing Song, May 16 2022

Formula

a(n) = [x^n] ((2 - x)*x*(x + 1))/(x^2 + x - 1)^2.

a(n) = (((-1 - sqrt(5))^(-n)*(sqrt(5)*n - n - 2) + (-1 + sqrt(5))^(-n)*(sqrt(5)*n + n + 2)))/(2^(1 - n)*sqrt(5)).

a(n) = (-1)^(n - 1)*(Fibonacci(-n) - n*Fibonacci(-n - 1)).

a(n) = (-1)^(n - 1)*A353595(-n, -n). (A353595 is defined for all n in Z.)

a(n) = ((-42*n^2 + 259*n - 350)*a(n - 3) + (123*n^2 - 76*n - 446)*a(n - 2) + (207*n^2 - 885*n + 412)*a(n - 1)) / ((165*n - 542)*(n - 1)) for n >= 4.

a(n) = Fibonacci(n) + n*Fibonacci(n+1). - Jianing Song, May 16 2022