A122491 a(n) = n * Fibonacci(n) - Sum_{i=0..n} Fibonacci(i).

0, 0, 0, 2, 5, 13, 28, 58, 114, 218, 407, 747, 1352, 2420, 4292, 7554, 13209, 22969, 39748, 68494, 117590, 201210, 343275, 584087, 991440, 1679208, 2838408, 4789058, 8066669, 13566373, 22782892, 38209762, 64003002, 107083610, 178967807, 298803459, 498404504

Offset: 0

Author

Ben Paul Thurston, Sep 16 2006

Keywords

• nonn
• easy

Comments

Similar to A190062.

Also the circuit rank and corank of the n-Lucas cube graph. - Eric W. Weisstein, Jul 28 2023

Examples

			a(5) = 13 because Fib(5) = 5, times 5 = 25 and subtract sum(Fib(5)) = 12 to get 13.
		

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.
• Eric Weisstein's World of Mathematics, Circuit Rank
• Eric Weisstein's World of Mathematics, Corank
• Eric Weisstein's World of Mathematics, Lucas Cube Graph
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Crossrefs

Cf. A000045.

Programs

Maple

with(combinat, fibonacci): for i from 1 to 30 do i*fibonacci(i) - sum(fibonacci(k), k=0..i); end do;

Mathematica

Table[n Fibonacci[n] - Fibonacci[n + 2] + 1, {n, 0, 40}] (* Stefan Steinerberger, Feb 22 2008 *)
LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 0, 2, 5}, 40] (* Harvey P. Dale, May 17 2016 *)

PARI

a(n)=n*fibonacci(n) - fibonacci(n+2) + 1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = n * Fibonacci(n) - Fibonacci(n+2) + 1. - Stefan Steinerberger, Feb 22 2008

G.f.: x^3*(2-x)/((1-x)*(1-x-x^2)^2). - Colin Barker, Feb 10 2012

a(n+2) = Sum_{k=0..n} A099920(k). - J. M. Bergot, Apr 13 2013

a(n) = 2*A006478(n)-A006478(n-1). - R. J. Mathar, May 04 2014

Extensions

Edited by N. J. A. Sloane, Sep 17 2006