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).
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
KEYWORD
nonn,easy
AUTHOR
Marcello M. Herreshoff (m(AT)marcello.gotdns.com), Jul 18 2006
STATUS
approved