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

0,3

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).

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).

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A287898 A129770 A134396 * A291006 A289781 A209239

Adjacent sequences: A059499 A059500 A059501 * A059503 A059504 A059505

Keyword

easy,nonn,nice,changed

Author

Floor van Lamoen, Jan 19 2001

Status

approved