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A120940
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Alternating sum of the Fibonacci numbers multiplied by their (combinatorial) indices.
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1
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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; internal format)
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OFFSET
| 0,3
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LINKS
| 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 to sequences with linear recurrences with constant coefficients, signature (1,3,-1,-3,-1).
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FORMULA
| a(n) = sum from k=0 to n of (-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
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MATHEMATICA
| CoefficientList[Series[(2*z^2 + z)/((z + 1)*(z^2 + z - 1)^2), {z, 0, 100}], z] (* From Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
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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)))))
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CROSSREFS
| Cf. A094584, A000045.
Sequence in context: A166212 A002219 A006906 * A049940 A051749 A030012
Adjacent sequences: A120937 A120938 A120939 * A120941 A120942 A120943
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KEYWORD
| nonn
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AUTHOR
| Marcello M. Herreshoff (m(AT)marcello.gotdns.com), Jul 18 2006
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