A277262 -id:A277262 - cp's OEIS frontend

A048116 a(n) = T(2n,n), where T is given by A048113.

1, 2, 4, 12, 36, 120, 408, 1440, 5160, 18816, 69336, 258048, 967344, 3649536, 13839504, 52715952, 201556944, 773182608, 2974442112, 11471570352, 44341125312, 171732665520, 666302137056, 2589317125824, 10076939895984, 39268487472336, 153208051192848

Offset: 1

Clark Kimberling

Number of planar walks starting at (1,1), ending at (n,n), remaining in the first quadrant and using steps (-1,2) and (2,-1).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Bousquet-Mélou and M. Petkovsek, Walks confined in a quadrant are not always D-finite

Cf. A048113, A277262.

b:= proc(l) option remember; `if`(l=[1$2], 1, add((p->
      `if`(p[1]<0, 0, b(p)))(sort(l-x)), x=[[-1, 2], [2, -1]]))
    end:
a:= n-> b([n$2]):
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 06 2016

b[l_List] := b[l] = If[l == {1, 1}, 1, Sum[Function[p, If[p[[1]] < 0, 0, b[p]]][Sort[l - x]], {x, {{-1, 2}, {2, -1}}}]]; a[n_] := b[{n, n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

a(n) ~ c * 4^n / sqrt(n), where c = 0.03748220353529780423030694970938451772844604409392271... . - Vaclav Kotesovec, Oct 07 2016

a(1)=1 prepended by Alois P. Heinz, Oct 06 2016

A328269 Number of walks on cubic lattice starting at (0,0,0), ending at (0,n,n), remaining in the first (nonnegative) octant and using steps (0,0,1), (0,1,0), (1,0,0), (-1,1,1), (1,-1,1), and (1,1,-1).

Original entry on oeis.org

1, 3, 26, 343, 5594, 103730, 2094028, 44889351, 1006126370, 23337166962, 556199376622, 13550764116530, 336190200180652, 8468872074477060, 216120719672921820, 5577150906683145103, 145324963753397617230, 3819107708757101038562, 101122686499165125017886

Offset: 0

Alois P. Heinz, Oct 10 2019

			a(1) = 3: [(0,0,0),(1,0,0),(0,1,1)], [(0,0,0),(0,1,0),(0,1,1)], [(0,0,0),(0,0,1),(0,1,1)].
a(2) = 26: [(0,0,0),(1,0,0),(2,0,0),(1,1,1),(0,2,2)], [(0,0,0),(1,0,0),(1,1,0),(1,1,1),(0,2,2)], ..., [(0,0,0),(0,0,1),(0,1,1),(0,1,2),(0,2,2)], [(0,0,0),(0,0,1),(0,0,2),(0,1,2),(0,2,2)].

Alois P. Heinz, Table of n, a(n) for n = 0..639
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Bisection (even part) of A328280.

Cf. A000225, A000984, A277262, A328270, A328300.

b:= proc(l) option remember; `if`(l[-1]=0, 1, (r-> add(
      add(add(`if`(i+j+k=1, (h-> `if`(h[1]<0, 0, b(h)))(
      sort(l-[i, j, k])), 0), k=r), j=r), i=r))([$-1..1]))
    end:
a:= n-> b([0, n$2]):
seq(a(n), n=0..23);

b[l_] := b[l] = If[Last[l] == 0, 1, Sum[If[i + j + k == 1, Function[h, If[h[[1]] < 0, 0, b[h]]][Sort[l - {i, j, k}]], 0], {i, {-1, 0, 1}}, {j, {-1, 0, 1}}, {k, {-1, 0, 1}}]];
a[n_] := b[{0, n, n}];
a /@ Range[0, 23] (* Jean-François Alcover, May 12 2020, after Maple *)

a(n) = A328300(2n,n).
a(n) is odd <=> n in { A000225 }.
a(n) ~ c * 2^(3*n) * (2 + sqrt(3))^n / n^2, where c =
0.081957778985952080274457799679795068000445171394180053136120884510526907545... - Vaclav Kotesovec, May 10 2020

cp's OEIS Frontend

A048116 a(n) = T(2n,n), where T is given by A048113.

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