|
%I #15 Apr 03 2019 10:47:58
%S 0,1,3,6,14,26,52,95,177,318,572,1012,1784,3117,5423,9382,16170,27758,
%T 47500,81035,137885,234046,396408,670056,1130544,1904281,3202587,
%U 5378310,9020102,15109058,25279012,42248567,70537929,117657342,196076468,326485852
%N Alternating sum of the Fibonacci numbers multiplied by their (combinatorial) indices.
%H Colin Barker, <a href="/A120940/b120940.txt">Table of n, a(n) for n = 0..1000</a>
%H M. M. Herreshoff, <a href="http://marcello.gotdns.com/~m/uncount.1.2.pdf">A Combinatorial proof of the summation from k = 0 to n of k times f sub k</a>, Presented at The Twelfth International Conference on Fibonacci Numbers and Their Applications.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,-3,-1).
%F 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).
%F a(n) = (-1)^n+A000045(n)-A001629(n+2)-3*A001629(n+1). - _R. J. Mathar_, Jul 11 2011
%F From _Colin Barker_, Apr 03 2019: (Start)
%F G.f.: x*(1 + 2*x) / ((1 + x)*(1 - x - x^2)^2).
%F a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 3*a(n-4) - a(n-5) for n>4.
%F (End)
%t CoefficientList[Series[(2*z^2 + z)/((z + 1)*(z^2 + z - 1)^2), {z, 0, 100}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2011 *)
%t LinearRecurrence[{1,3,-1,-3,-1},{0,1,3,6,14},40] (* _Harvey P. Dale_, Apr 21 2018 *)
%o #!/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)))))
%o (PARI) concat(0, Vec(x*(1 + 2*x) / ((1 + x)*(1 - x - x^2)^2) + O(x^40))) \\ _Colin Barker_, Apr 03 2019
%Y Cf. A094584, A000045.
%K nonn,easy
%O 0,3
%A Marcello M. Herreshoff (m(AT)marcello.gotdns.com), Jul 18 2006
|
