A264147 a(n) = n*F(n+1) - (n+1)*F(n), where F = A000045.

0, -1, 1, 1, 5, 10, 22, 43, 83, 155, 285, 516, 924, 1639, 2885, 5045, 8773, 15182, 26162, 44915, 76855, 131119, 223101, 378696, 641400, 1084175, 1829257, 3081193, 5181893, 8702290, 14594830, 24446971, 40902299, 68359619, 114132765, 190373580, 317258388, 528265207

Offset: 0

Author

Bruno Berselli, Nov 04 2015

Keywords

• sign
• easy

Comments

a(n) is prime for n = 4, 7, 8, 26, 28, 52, 86, 87, 93, 97, 158, 196, 303, 2908, 3412, 4111, 4208, 6183, 6337, 9878, ...

Links

• Bruno Berselli, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

Cf. A000045, A001629, A007502, A354044.

Cf. A178521: n*F(n+1) + (n+1)*F(n).

Cf. A094588: n*F(n-1) + F(n).

Cf. A099920: Sum_{i=0..n} F(i)*L(n-i).

Cf. A023607: Sum_{i=0..n} F(i)*L(n+1-i).

Programs

Julia

# The function 'fibrec' is defined in A354044.
function A264147(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n)
    n*b - a*(n + 1)
end # Peter Luschny, May 16 2022

Magma

[n*Fibonacci(n+1)-(n+1)*Fibonacci(n): n in [0..40]];

Maple

A264147 := proc(n)
    n*combinat[fibonacci](n+1)-(n+1)*combinat[fibonacci](n) ;
end proc:
seq(A264147(n),n=0..10) ; # R. J. Mathar, Jun 02 2022

Mathematica

Table[n Fibonacci[n + 1] - (n + 1) Fibonacci[n], {n, 0, 40}]

Maxima

makelist(n*fib(n+1)-(n+1)*fib(n), n, 0, 40);

PARI

for(n=0, 40, print1(n*fibonacci(n+1)-(n+1)*fibonacci(n)", "));

PARI

concat(0, Vec(-x*(1 - 3*x) / (1 - x - x^2)^2 + O(x^50))) \\ Colin Barker, Jul 27 2017

Sage

[n*fibonacci(n+1)-(n+1)*fibonacci(n) for n in (0..40)]

Formula

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

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

a(n) = n*F(n-1) - F(n).

a(n) = Sum_{i=0..n} F(i)*L(n-1-i), where L() is a Lucas number (A000032).

a(n) = 3*A001629(n) - A001629(n+1).

a(n) = -(-1)^n*A178521(-n).

a(n+2) - a(n) = A007502(n+1).

Sum_{i>0} 1/a(i) = 1.39516607051636028893879220294180374...

a(n) = (-((1+sqrt(5))/2)^n*(2*sqrt(5) + (-5+sqrt(5))*n) + ((1-sqrt(5))/2)^n*(2*sqrt(5) + (5+sqrt(5))*n)) / 10. - Colin Barker, Jul 27 2017

a(n) = (-i)^n*(n*sin(c*(n+1)) - (n+1)*sin(c*n)*i)/sqrt(5/4) where c = arccos(i/2). - Peter Luschny, May 16 2022