A120940 Alternating sum of the Fibonacci numbers multiplied by their (combinatorial) indices.

0, 1, 3, 6, 14, 26, 52, 95, 177, 318, 572, 1012, 1784, 3117, 5423, 9382, 16170, 27758, 47500, 81035, 137885, 234046, 396408, 670056, 1130544, 1904281, 3202587, 5378310, 9020102, 15109058, 25279012, 42248567, 70537929, 117657342, 196076468, 326485852

Offset

0,3

Links

Colin Barker, Table of n, a(n) for n = 0..1000

M. M. Herreshoff, A Combinatorial proof of the summation from k = 0 to n of k times f sub k, Presented at The Twelfth International Conference on Fibonacci Numbers and Their Applications.

Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-3,-1).

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*k*f(k) also, when n >= 3, a(n) = nf(n-1) + f(n-3) + (-1)^n where f(n) = F(n+1).

a(n) = (-1)^n+A000045(n)-A001629(n+2)-3*A001629(n+1). - R. J. Mathar, Jul 11 2011

From Colin Barker, Apr 03 2019: (Start)

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

a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 3*a(n-4) - a(n-5) for n>4.

(End)

Mathematica

CoefficientList[Series[(2*z^2 + z)/((z + 1)*(z^2 + z - 1)^2), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
LinearRecurrence[{1, 3, -1, -3, -1}, {0, 1, 3, 6, 14}, 40] (* Harvey P. Dale, Apr 21 2018 *)

Prog

#!/usr/bin/guile -s Computes the alternating sum of the fibonacci numbers multiplied by their (combinatorial) indices. !# (use-modules (srfi srfi-1)) (define (fibo n) (define (iter a b k) (if (= k n) b (iter b (+ a b) (+ k 1)))) (iter 0 1 0)) (define (a n) (fold + 0 (map (lambda (k) (* k (fibo k) (expt -1 (- n k)))) (iota (+ n 1)))))
(PARI) concat(0, Vec(x*(1 + 2*x) / ((1 + x)*(1 - x - x^2)^2) + O(x^40))) \\ Colin Barker, Apr 03 2019

Crossrefs

Cf. A094584, A000045.

Sequence in context: A002219 A006906 A324703 * A049940 A265947 A358831

Adjacent sequences: A120937 A120938 A120939 * A120941 A120942 A120943

Keyword

nonn,easy

Author

Marcello M. Herreshoff (m(AT)marcello.gotdns.com), Jul 18 2006

Status

approved