A223718 Number of unimodal functions [1..n]->[0..2].

1, 3, 9, 22, 46, 86, 148, 239, 367, 541, 771, 1068, 1444, 1912, 2486, 3181, 4013, 4999, 6157, 7506, 9066, 10858, 12904, 15227, 17851, 20801, 24103, 27784, 31872, 36396, 41386, 46873, 52889, 59467, 66641, 74446, 82918, 92094, 102012, 112711, 124231

Offset: 0

Author

R. H. Hardin, Mar 26 2013

Keywords

• nonn
• easy

Comments

Column 1 of A223725.

Examples

			Some solutions for n=3
..1....2....0....1....0....2....1....2....0....2....0....1....0....1....0....1
..2....1....1....1....0....0....0....1....0....2....2....1....1....2....2....2
..0....1....0....0....1....0....0....0....2....2....1....1....1....2....0....1
From _Joerg Arndt_, Dec 27 2023: (Start)
The a(3) = 22 such functions are (dots for zeros)
   1:  [ . . . ]
   2:  [ . . 1 ]
   3:  [ . . 2 ]
   4:  [ . 1 . ]
   5:  [ . 1 1 ]
   6:  [ . 1 2 ]
   7:  [ . 2 . ]
   8:  [ . 2 1 ]
   9:  [ . 2 2 ]
  10:  [ 1 . . ]
  11:  [ 1 1 . ]
  12:  [ 1 1 1 ]
  13:  [ 1 1 2 ]
  14:  [ 1 2 . ]
  15:  [ 1 2 1 ]
  16:  [ 1 2 2 ]
  17:  [ 2 . . ]
  18:  [ 2 1 . ]
  19:  [ 2 1 1 ]
  20:  [ 2 2 . ]
  21:  [ 2 2 1 ]
  22:  [ 2 2 2 ]
(End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..210 from R. H. Hardin)
• Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Crossrefs

Column m=3 of A071920.

Cf. A000124 (unimodal functions [1..n]->[0..1]), A088536 ([1..n] -> [1..n]).

Formula

a(n) = A071920(n,3) = 1+n*(n+1)*(n^2+5*n+18)/24.

G.f.: 1-x*(x^2-3*x+3)*(x^2-x+1) / (x-1)^5 . a(n) = 1+A051744(n). - R. J. Mathar, May 17 2014

Extensions

a(0)=1 prepended by Alois P. Heinz, Dec 27 2023