A276029 Number of ways to transform a sequence of n ones and n twos to a single number by continually removing two numbers and replacing them with their sum modulo 3.

1, 4, 27, 228, 2226, 23778, 270693, 3229106, 39922172, 507680620, 6604676830, 87549425068, 1178880306174, 16086844260290, 222045139578443, 3095457073064120, 43529719213465854, 616853383573066504, 8801227720060618544, 126344910516550743232

Offset

1,2

Comments

Originally this entry had a reference to a paper on the arXiv by Caleb Ji, Enumerative Properties of Posets Corresponding to a Certain Class of No Strategy Games, arXiv:1608.06025 [math.CO], 2016. However, this article has since been removed from the arXiv. - N. J. A. Sloane, Sep 07 2018

Links

Alois P. Heinz, Table of n, a(n) for n = 1..800

Formula

a(n) = b(0, n, n) where f(a, b, c) is the number of ways to reach one number beginning with a zeros, b ones, and c twos.

Then f(a, b, c) = f_1 + f_2 + f_3 + f_4 where f_1 = f(a-1, b, c) if a>=2 or a, b >=1 or a,c >=1, f_2 = f(a, b-2, c+1) if b >= 2, f_3 = f(a, b+1, c-2) if c >= 2, and f_4 = f(a+1, b-1, c-1) if b, c >= 1, and each are 0 otherwise. Initial terms: f(a, b, c) = 1 for all 1 <= a+b+c <= 2, where a, b, c >= 0.

Maple

b:= proc(x, y, z) option remember;
      `if`(x+y+z=1, 1, `if`(y>0 and z>0, b(x+1, y-1, z-1), 0)+
      `if`(x>1 or x>0 and y>0 or x>0 and z>0, b(x-1, y, z), 0)+
      `if`(y>1, b(x, y-2, z+1), 0)+`if`(z>1, b(x, y+1, z-2), 0))
    end:
a:= n-> b(0, n, n):
seq(a(n), n=1..35);  # Alois P. Heinz, Aug 18 2016

Mathematica

b[x_, y_, z_] := b[x, y, z] = If[x + y + z == 1, 1, If[y > 0 && z > 0, b[x + 1, y - 1, z - 1], 0] + If[x > 1 || x > 0 && y > 0 || x > 0 && z > 0, b[x - 1, y, z], 0] + If[y > 1, b[x, y - 2, z + 1], 0] + If[z > 1, b[x, y + 1, z - 2], 0]];
a[n_] := b[0, n, n];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)

Crossrefs

Cf. A276027, A276028.

Sequence in context: A317103 A341962 A354588 * A160883 A362274 A328978

Adjacent sequences: A276026 A276027 A276028 * A276030 A276031 A276032

Keyword

nonn

Author

Caleb Ji, Aug 17 2016

Extensions

More terms from Alois P. Heinz, Aug 18 2016

Status

approved