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A045925 a(n) = n*Fibonacci(n). 22
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 (list; graph; refs; listen; history; text; internal format)
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

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

S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.

Index to divisibility sequences

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(binomial(n-k,k)/(n-k), k=0..floor(n/2)). - Gary Detlefs, Jan 19 2013

a(n) = A014965(n) * A104714(n). - Michel Marcus, Oct 24 2013

a(n) = 3*A001629(n+1) - A001629(n+2) + A000045(n-1). - Ralf Stephan, Apr 26 2014

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^4-2x^3-x^2+2x+1. R. K. Guy, Nov 13 2015

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

Partial sums: A014286. Cf. A000045.

Cf. A099920, A023607.

Sequence in context: A137829 A262196 A261667 * A128020 A116562 A140659

Adjacent sequences:  A045922 A045923 A045924 * A045926 A045927 A045928

KEYWORD

nonn,easy

AUTHOR

Jeff Burch

EXTENSIONS

Incorrect formula removed by Gary Detlefs, Oct 27 2011

STATUS

approved

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Last modified November 18 02:54 EST 2017. Contains 294840 sequences.