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

Author

Caleb Ji, Aug 17 2016

Keywords

• nonn

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

Crossrefs

Cf. A276027, A276028.

Programs

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 *)

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.

Extensions

More terms from Alois P. Heinz, Aug 18 2016