A045925 a(n) = n*Fibonacci(n).
0, 1, 2, 6, 12, 25, 48, 91, 168, 306, 550, 979, 1728, 3029, 5278, 9150, 15792, 27149, 46512, 79439, 135300, 229866, 389642, 659111, 1112832, 1875625, 3156218, 5303286, 8898708, 14912641, 24961200, 41734339, 69705888, 116311074, 193898158, 322961275, 537492672
Offset: 0
References
- Jean Paul Van Bendegem, The Heterogeneity of Mathematical Research, a chapter in Perspectives on Interrogative Models of Inquiry, Volume 8 of the series Logic, Argumentation & Reasoning pp 73-94, Springer 2015. See Section 2.1.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- Russell Euler, Problem B-670, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 28, No. 3 (1990), p. 277; Application of Generating Functions, Solution to Problem B-670 by Russell Jay Hendel, ibid., Vol. 29, No. 3 (1991), p. 278.
- Rigoberto Flórez, Robinson Higuita and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
- Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
- Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics, Vol. 26, No. 3 (2019), Article P3.26.
- Silvia Heubach and Toufik Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
- A. G. Shannon, B. Kuloğlu, and E. Özkan, Rhaly terraced sequences their generalizations, properties and applications, Comp. Appl. Math. 44, 226 (2025). See p. 2.
- Kai Ting Keshia Yap, David Wehlau and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
- Index to divisibility sequences.
- Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).
Programs
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Haskell
a045925 n = a045925_list !! (n-1) a045925_list = zipWith (*) [0..] a000045_list -- Reinhard Zumkeller, Oct 01 2012
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Magma
[n*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011
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Maple
a:= n-> n*(<<0|1>, <1|1>>^n)[1,2]: seq(a(n), n=0..37); # Alois P. Heinz, May 07 2021
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Mathematica
Table[Fibonacci[n]*n, {n, 0, 33}] (* Zerinvary Lajos, Jul 09 2009 *) LinearRecurrence[{2, 1, -2, -1}, {0, 1, 2, 6}, 34] (* or *) CoefficientList[ Series[(x + x^3)/(-1 + x + x^2)^2, {x, 0, 35}], x] (* Robert G. Wilson v, Nov 14 2015 *) -
PARI
Lucas(n)=fibonacci(n-1)+fibonacci(n+1) a(n)=polcoeff(sum(m=1,n,eulerphi(m)*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n) \\ Paul D. Hanna, Jan 12 2012
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PARI
a(n)=n*fibonacci(n) \\ Charles R Greathouse IV, Jan 12 2012
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PARI
concat(0, Vec(x*(1+x^2)/(1-x-x^2)^2 + O(x^100))) \\ Altug Alkan, Oct 28 2015
Formula
G.f.: x*(1+x^2)/(1-x-x^2)^2.
G.f.: Sum_{n>=1} phi(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Fibonacci(n)*x^n, where phi(n) = A000010(n) and Lucas(n) = A000204(n). - Paul D. Hanna, Jan 12 2012
a(n) = a(n-1) + a(n-2) + L(n-1). - Gary Detlefs, Dec 29 2012
a(n) = F(n+1) + Sum_{k=1..n-2} F(k)*L(n-k), F = A000045 and L = A000032. - Gary Detlefs, Dec 29 2012
a(n) = F(2*n)/Sum_{k=0..floor(n/2)} binomial(n-k,k)/(n-k). - Gary Detlefs, Jan 19 2013
a(n) = 2*n*(F(n-2) + floor(F(n-3)/2)) + (n^3 mod 3*n), F = A000045. - Gary Detlefs, Jun 06 2014
E.g.f.: x*(exp(-x/phi)/phi + exp(x*phi)*phi)/sqrt(5), where phi = (1+sqrt(5))/2. - Vladimir Reshetnikov, Oct 28 2015
This is a divisibility sequence and is generated by x^4 - 2*x^3 - x^2 + 2*x + 1. - R. K. Guy, Nov 13 2015
a(n) = L'(n, 1), the first derivative of the n-th Lucas polynomial evaluated at 1. - Andrés Ventas, Nov 12 2021
Sum_{n>=0} a(n)/2^n = 10 (Euler, 1990). - Amiram Eldar, Jan 22 2022
Extensions
Incorrect formula removed by Gary Detlefs, Oct 27 2011
Comments