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A120940 Alternating sum of the Fibonacci numbers multiplied by their (combinatorial) indices. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A323450

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

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Last modified September 20 12:43 EDT 2019. Contains 327238 sequences. (Running on oeis4.)