A228248 Number of 2n-step lattice paths from (0,0) to (0,0) using steps in {N, S, E, W} starting with East, then always moving straight ahead or turning left.

1, 0, 1, 3, 9, 30, 103, 357, 1257, 4494, 16246, 59246, 217719, 805389, 2996113, 11200113, 42047593, 158452138, 599113966, 2272065638, 8639763574, 32933685102, 125817012366, 481631387438, 1847110931703, 7095928565405, 27302745922817, 105204285608025

Offset

0,4

Comments

From Gus Wiseman, Oct 13 2022: (Start)

Also the number of integer compositions of 2n whose half-alternating and skew-alternating sums are both 0, where we define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ..., and the skew-alternating sum to be A - B - C + D + E - F - G + ... For example, the a(0) = 1 through a(4) = 9 compositions are:

() . (1111) (1212) (1313)

(2121) (2222)

(11211) (3131)

(11312)

(12221)

(21311)

(112211)

(1112111)

(11111111)

For skew-alternating only: A001700, ranked by A357627, reverse A357628.

For partitions: A035363, half only A357639, skew only A357640.

For half-alternating only: A357641, ranked by A357625, reverse A357626.

These compositions are ranked by A357706.

(End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

David Scambler et al., A new lattice walk and follow-up messages on the SeqFan list, May 08 2013.

Formula

a(n) ~ 2^(2n-1)/(Pi*n). - Vaclav Kotesovec, Jul 16 2014

Example

a(0) = 1: [], the empty path.
a(1) = 0.
a(2) = 1: ENWS.
a(3) = 3: EENWWS, ENNWSS, ENWWSE.

Maple

b:= proc(x, y, n) option remember; `if`(abs(x)+abs(y)>n, 0,
      `if`(n=0, 1, b(x+1, y, n-1) +b(y+1, -x, n-1)))
    end:
a:= n-> ceil(b(0, 0, 2*n)/2):
seq(a(n), n=0..40);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [1, 0, 1, 3, 9][n+1],
     ((n-1)*(414288-1901580*n+186029*n^6-869551*n^5+2393807*n^4
     -3938624*n^3+3753546*n^2+1050*n^8-21605*n^7)*a(n-1)
     +(-17751540*n-12215020*n^5+3494038*n^6+3777840+27478070*n^4
     -39711374*n^3+35488098*n^2-2700*n^9+62370*n^8-621126*n^7)*a(n-2)
     +(-18193248+77490792*n-9138800*n^6+35323128*n^5-88122332*n^4
     +141370392*n^3-140075264*n^2+5400*n^9-135540*n^8+1476432*n^7)*a(n-3)
     +(-192473328*n+48577536+17091500*n^6-70036368*n^5+184890672*n^4
     -313388816*n^3+328043052*n^2-8400*n^9+224440*n^8-2600032*n^7)*a(n-4)
     +8*(n-5)*(150*n^6-2015*n^5+10852*n^4-29867*n^3+44208*n^2-33540*n
     +10416)*(-9+2*n)^2*a(n-5)) / (n^2*(396988*n-487261*n^2+150*n^7
     -3065*n^6+26092*n^5-119602*n^4+317746*n^3-131048)))
    end:
seq(a(n), n=0..40);

Mathematica

b[x_, y_, n_] := b[x, y, n] = If[Abs[x] + Abs[y] > n, 0, If[n == 0, 1, b[x + 1, y, n - 1] + b[y + 1, -x, n - 1]]];
a[n_] := Ceiling[b[0, 0, 2n]/2];
a /@ Range[0, 40] (* Jean-François Alcover, May 14 2020, after Maple *)
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], halfats[#]==0&&skats[#]==0&]], {n, 0, 7}] (* Gus Wiseman, Oct 12 2022 *)

Crossrefs

Cf. A004006 (same rules, but self-avoiding).

Cf. A000583, A001511, A001700, A088218, A357136, A357182, A357642.

Sequence in context: A148949 A148950 A148951 * A047064 A047007 A007453

Adjacent sequences: A228245 A228246 A228247 * A228249 A228250 A228251

Keyword

nonn,easy

Author

David Scambler and Alois P. Heinz, Aug 18 2013

Status

approved