A243965 -id:A243965 - cp's OEIS frontend

A243820 Number of Dyck paths of semilength n such that all sixteen consecutive step patterns of length 4 occur at least once.

38, 587, 4785, 31398, 190050, 1043248, 5324534, 25711105, 119092876, 533680433, 2329450085, 9955122396, 41824314441, 173289259905, 709861015186, 2880803895035, 11601285215222, 46422795985447, 184784743066842, 732324944072523, 2891815190097065, 11385122145001833

Offset: 10

Alois P. Heinz, Jun 11 2014

nonn

			a(10) = 38: 10101100111101000010, 10101101001111000010, 10101111000011010010, 10101111001101000010, 10101111010000110010, 10101111010011000010, 10110011110100001010, 10110011110101000010, 10110100111100001010, 10110101001111000010, 10111100001101001010, 10111100001101010010, 10111100110100001010, 10111100110101000010, 10111101000011001010, 10111101001100001010, 10111101010000110010, 10111101010011000010, 11001011110000110100, 11001011110100001100, 11001101011110000100, 11001101111000010100, 11001111000010110100, 11001111010000101100, 11001111010110000100, 11001111011000010100, 11010010111100001100, 11010011110000101100, 11010110011110000100, 11010111100001001100, 11010111100001100100, 11010111100100001100, 11010111100110000100, 11011001111000010100, 11011110000101001100, 11011110000110010100, 11011110010100001100, 11011110011000010100.  Here 1=Up=(1,1), 0=Down=(1,-1).

Alois P. Heinz, Table of n, a(n) for n = 10..70

Cf. A242257, A243882, A243965, A243966.

b:= proc(x, y, t, s) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, `if`(s={}, 1, 0), `if`(nops(s)>x, 0, add(
      b(x-1, y-1+2*j, irem(2*t+j, 8), s minus {2*t+j}), j=0..1))))
    end:
a:= n-> add(b(2*n-3, l[], {$0..15}), l=[[1, 5], [1, 6], [3, 7]]):
seq(a(n), n=10..20);

b[x_, y_, t_, s_List] := b[x, y, t, s] = If[y < 0 || y > x, 0, If[x == 0, If[s == {}, 1, 0], If[Length[s] > x, 0, Sum[b[x - 1, y - 1 + 2 j, Mod[2 t + j, 8], s  ~Complement~  {2 t + j}], {j, 0, 1}]]]];
a[n_] := Sum[b[2 n - 3, Sequence @@ l, Range[0, 15]], {l, {{1, 5}, {1, 6}, {3, 7}}}];
a /@ Range[10, 31] (* Jean-François Alcover, Apr 28 2020, after Alois P. Heinz *)

A243966 Number of Dyck paths of semilength n such that all five consecutive patterns of Dyck paths of semilength 3 occur at least once.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 138, 1152, 8166, 52098, 308964, 1733444, 9311300, 48280464, 243112106, 1194286106, 5744306228, 27129749648, 126111332862, 578106334718, 2617667137358, 11723920607410, 51998857149406, 228621028644376, 997286152915772

Offset: 0

Alois P. Heinz, Jun 16 2014

nonn

The five consecutive patterns that occur at least once each are 101010, 101100, 110010, 110100, 111000. Here 1=Up=(1,1), 0=Down=(1,-1).

			a(12) = 12: 101010110010110100111000, 101010110010111000110100, 101100101010110100111000, 101100101010111000110100, 110100101010110010111000, 110100101100101010111000, 110100111000101010110010, 110100111000101100101010, 111000101010110010110100, 111000101100101010110100, 111000110100101010110010, 111000110100101100101010.
Here 1=Up=(1,1), 0=Down=(1,-1).

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A014486, A063171, A243820, A243965, A243986.

b:= proc(x, y, l) option remember; local m; m:= min(l[]);
      `if`(y>x or y<0 or 7-m>x, 0, `if`(x=0, 1,
      b(x-1, y+1, [[2, 3, 4, 4, 2, 2, 7][l[1]],
       [2, 3, 3, 5, 3, 2, 7][l[2]], [2, 3, 3, 2, 6, 3,7][l[3]],
       [2, 2, 4, 5, 2, 4, 7][l[4]], [2, 2, 4, 2, 6, 2,7][l[5]]])+
      b(x-1, y-1, [[1, 1, 1, 5, 6, 7, 7][l[1]],
       [1, 1, 4, 1, 6, 7, 7][l[2]], [1, 1, 4, 5, 1, 7, 7][l[3]],
       [1, 3, 1, 3, 6, 7, 7][l[4]], [1, 3, 1, 5, 1, 7, 7][l[5]]])))
    end:
a:= n-> b(2*n, 0, [1$5]):
seq(a(n), n=0..35);

b[x_, y_, l_] := b[x, y, l] = Module[{m = Min[l]},
        If[y>x || y<0 || 7-m>x, 0, If[x == 0, 1,
        b[x-1, y+1, MapIndexed[#1[[l[[#2[[1]] ]] ]]&,
        {{2, 3, 4, 4, 2, 2, 7},
         {2, 3, 3, 5, 3, 2, 7},
         {2, 3, 3, 2, 6, 3, 7},
         {2, 2, 4, 5, 2, 4, 7},
         {2, 2, 4, 2, 6, 2, 7}}]]] +
        b[x-1, y-1, MapIndexed[#1[[l[[#2[[1]] ]] ]]&,
        {{1, 1, 1, 5, 6, 7, 7},
         {1, 1, 4, 1, 6, 7, 7},
         {1, 1, 4, 5, 1, 7, 7},
         {1, 3, 1, 3, 6, 7, 7},
         {1, 3, 1, 5, 1, 7, 7}}]]]];
a[n_] := b[2n, 0, {1, 1, 1, 1, 1}];
a /@ Range[0, 35] (* Jean-François Alcover, Jan 26 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A243820 Number of Dyck paths of semilength n such that all sixteen consecutive step patterns of length 4 occur at least once.

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