OFFSET
0,3
COMMENTS
Number of levels in all compositions of n+1 with only 1's and 2's.
Apart from first term: row sums of the triangle in A131410. - Reinhard Zumkeller, Oct 07 2012
Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the third one. - Sergey Kitaev, Dec 08 2020
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 entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).
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
MAPLE
a:= n-> n*(<<0|1>, <1|1>>^n)[1, 2]:
seq(a(n), n=0..37); # Alois P. Heinz, May 07 2021
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 *)
PROG
(Magma) [n*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011
(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
(PARI) a(n)=n*fibonacci(n) \\ Charles R Greathouse IV, Jan 12 2012
(PARI) concat(0, Vec(x*(1+x^2)/(1-x-x^2)^2 + O(x^100))) \\ Altug Alkan, Oct 28 2015
(Haskell)
a045925 n = a045925_list !! (n-1)
a045925_list = zipWith (*) [0..] a000045_list
-- Reinhard Zumkeller, Oct 01 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Incorrect formula removed by Gary Detlefs, Oct 27 2011
STATUS
approved