A202058 - OEIS

%I #43 Jan 19 2024 21:13:59

%S 1,1,2,4,10,27,83,277,1015,4007,17047,77451,374889,1923168,10427250,

%T 59544957,357236992,2245822801,14762969601,101264286082,723499803180,

%U 5375063821727,41459660565329,331546282841906,2745163969235517,23505333233440927,207895424692608432

%N Number of ascent sequences avoiding the pattern 000.

%C It appears that no formula or g.f. is known.

%H Andrew Conway and Miles Conway, <a href="/A202058/b202058.txt">Table of n, a(n) for n = 0..176</a>

%H Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, <a href="https://arxiv.org/abs/2111.01279">Pattern-avoiding ascent sequences of length 3</a>, arXiv:2111.01279 [math.CO], Nov 01 2021.

%H P. Duncan and Einar Steingrimsson, <a href="https://arxiv.org/abs/1109.3641">Pattern avoidance in ascent sequences</a>, arXiv preprint arXiv:1109.3641, 2011

%t b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n==0, 1, Sum[If[Coefficient[ p, x, j]==k, 0, b[n-1, j, t + If[j>i, 1, 0], p+x^j, k]], {j, 1, t+1}]];

%t a[n_] := b[n, 0, 0, 0, Min[n, 2]];

%t Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 17}] (* _Jean-François Alcover_, Sep 01 2018, after _Alois P. Heinz_ in A294220 *)

%Y Total number of ascent sequences is given by A022493. Number of ascent sequences avoiding 001 (and others) is A000079; 102 is A007051; 101 is A000108; 000 is A202058; 100 is A202059; 110 is A202060; 120 is A202061; 201 is A202062; 210 is A108304; 0123 is A080937; 0021 is A007317; 0000 is A317784.

%Y Column k=2 of A294220.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Dec 10 2011

%E a(15)-a(17) from _Alois P. Heinz_, Nov 09 2012

%E a(18)-a(20) from _Giovanni Resta_, Jan 06 2014

%E a(21) from _Vaclav Kotesovec_, Aug 21 2018

%E a(22) from _Vaclav Kotesovec_, Aug 22 2018

%E More terms from _Anthony Guttmann_, Nov 04 2021