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

%I A223718 #30 Dec 27 2023 16:31:27
%S A223718 1,3,9,22,46,86,148,239,367,541,771,1068,1444,1912,2486,3181,4013,
%T A223718 4999,6157,7506,9066,10858,12904,15227,17851,20801,24103,27784,31872,
%U A223718 36396,41386,46873,52889,59467,66641,74446,82918,92094,102012,112711,124231
%N A223718 Number of unimodal functions [1..n]->[0..2].
%C A223718 Column 1 of A223725.
%H A223718 Alois P. Heinz, <a href="/A223718/b223718.txt">Table of n, a(n) for n = 0..10000</a> (terms n=1..210 from R. H. Hardin)
%H A223718 Kyu-Hwan Lee and Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%F A223718 a(n) = A071920(n,3) = 1+n*(n+1)*(n^2+5*n+18)/24.
%F A223718 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
%e A223718 Some solutions for n=3
%e A223718 ..1....2....0....1....0....2....1....2....0....2....0....1....0....1....0....1
%e A223718 ..2....1....1....1....0....0....0....1....0....2....2....1....1....2....2....2
%e A223718 ..0....1....0....0....1....0....0....0....2....2....1....1....1....2....0....1
%e A223718 From _Joerg Arndt_, Dec 27 2023: (Start)
%e A223718 The a(3) = 22 such functions are (dots for zeros)
%e A223718    1:  [ . . . ]
%e A223718    2:  [ . . 1 ]
%e A223718    3:  [ . . 2 ]
%e A223718    4:  [ . 1 . ]
%e A223718    5:  [ . 1 1 ]
%e A223718    6:  [ . 1 2 ]
%e A223718    7:  [ . 2 . ]
%e A223718    8:  [ . 2 1 ]
%e A223718    9:  [ . 2 2 ]
%e A223718   10:  [ 1 . . ]
%e A223718   11:  [ 1 1 . ]
%e A223718   12:  [ 1 1 1 ]
%e A223718   13:  [ 1 1 2 ]
%e A223718   14:  [ 1 2 . ]
%e A223718   15:  [ 1 2 1 ]
%e A223718   16:  [ 1 2 2 ]
%e A223718   17:  [ 2 . . ]
%e A223718   18:  [ 2 1 . ]
%e A223718   19:  [ 2 1 1 ]
%e A223718   20:  [ 2 2 . ]
%e A223718   21:  [ 2 2 1 ]
%e A223718   22:  [ 2 2 2 ]
%e A223718 (End)
%Y A223718 Column m=3 of A071920.
%Y A223718 Cf. A000124 (unimodal functions [1..n]->[0..1]), A088536 ([1..n] -> [1..n]).
%K A223718 nonn,easy
%O A223718 0,2
%A A223718 _R. H. Hardin_, Mar 26 2013
%E A223718 a(0)=1 prepended by _Alois P. Heinz_, Dec 27 2023