A190062 a(n) = n*Fibonacci(n) - Sum_{i=0..n-1} Fibonacci(i).

0, 1, 1, 4, 8, 18, 36, 71, 135, 252, 462, 836, 1496, 2653, 4669, 8164, 14196, 24566, 42332, 72675, 124355, 212156, 360986, 612744, 1037808, 1754233, 2959801, 4985476, 8384480, 14080602, 23614932, 39556031, 66181311, 110608188, 184670694

Offset: 0

Author

Bruno Berselli, May 04 2011

Keywords

• nonn
• easy

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Carlos Alirio Rico Acevedo, Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Crossrefs

Cf. A122491, A045925, A000071.

Programs

Magma

[0] cat [n*Fibonacci(n)-(&+[Fibonacci(k): k in [0..n-1]]): n in [1..34]];

Mathematica

CoefficientList[Series[x (1 - 2 x + 2 x^2) / ((1 - x) (1 - x - x^2)^2), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 19 2013 *)

PARI

concat(0, Vec(x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2) + O(x^50))) \\ Altug Alkan, Nov 13 2015

Formula

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

a(n) = A045925(n) - A000071(n+1).

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

a(n) = (((2*n-1)*r-5)*(1+r)^n-((2*n-1)*r+5)*(1-r)^n)/(10*2^n)+1, where r=sqrt(5).