A059502 a(n) = (3*n*F(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.

0, 1, 3, 9, 27, 80, 234, 677, 1941, 5523, 15615, 43906, 122868, 342409, 950727, 2631165, 7260579, 19982612, 54865566, 150316973, 411015705, 1121818311, 3056773383, 8316416134, 22593883752, 61301547025, 166118284299, 449639574897, 1215751720491, 3283883157848

Offset: 0

Author

Floor van Lamoen, Jan 19 2001

Keywords

• easy
• nonn
• nice

Comments

Substituting x(1-x)/(1-2x) into x/(1-x)^2 yields g.f. of sequence.

Variation of A059216 (and of Boustrophedon transform) applied to 1,2,3,4,...: fill an array by diagonals, each time in the same direction, say the 'up' direction. The first column is 1,2,3,4,... For the next element of a diagonal, add to the previous element the elements of the row the new element is in. The first row gives a(n).

Examples

			The array (see A059503) begins
  1 3  9 27 80 ...
  2 5 14 40 ...
  3 7 19 ...
  4 9  5 ...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
• Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 17, 19.
• Index entries for two-way infinite sequences
• Index entries for sequences related to boustrophedon transform
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Crossrefs

Cf. A000045, A000667, A059216, A059219, A027994, A059512, A059503.

Programs

Magma

[(3*n*Fibonacci(2*n-1)+(3-n)*Fibonacci(2*n))/5: n in [0..100]]; // Vincenzo Librandi, Apr 23 2011

Mathematica

Table[(3n Fibonacci[2n-1]+(3-n)Fibonacci[2n])/5,{n,0,30}] (* or *) CoefficientList[Series[x(1-x)(1-2x)/(1-3x+x^2)^2,{x,0,30}],x] (* Harvey P. Dale, Apr 23 2011 *)

PARI

a(n)=(3*n*fibonacci(2*n-1)+(3-n)*fibonacci(2*n))/5

Formula

a(n) = 2*a(n-1) + Sum{m<=n-2} a(m) + A001519(n-2).

G.f.: x*(1 - x)*(1 - 2*x)/(1 - 3*x + x^2)^2. - Emeric Deutsch, Oct 07 2002

a(n) = A147703(n,1). - Philippe Deléham, Nov 29 2008

a(n) = A001871(n-1) - 3*A001871(n-2) + 2*A001871(n-3). - R. J. Mathar, Apr 09 2019

E.g.f.: 2*exp(3*x/2)*(5*x*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025