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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. 2
1, 4, 27, 228, 2226, 23778, 270693, 3229106, 39922172, 507680620, 6604676830, 87549425068, 1178880306174, 16086844260290, 222045139578443, 3095457073064120, 43529719213465854, 616853383573066504, 8801227720060618544, 126344910516550743232 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Nov 10 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A276027, A276028.

Sequence in context: A319518 A304045 A317103 * A160883 A185655 A181146

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

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Last modified October 21 08:00 EDT 2018. Contains 316405 sequences. (Running on oeis4.)