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A276028 Number of ways to transform a sequence of n zeros and n ones to a single number by continually removing two numbers and replacing them with their sum modulo 3. 2
1, 3, 10, 50, 259, 1540, 9594, 62649, 422598, 2960716, 21030711, 152470357, 1129502128, 8434189996, 63674017174, 488573782216, 3762932025753, 29178861276815, 229208503750838, 1803350026315019, 14248236439629534, 113825380196996557, 909507867095014380 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also the number of distinct transformations when the initial state consists of n zeros and n twos.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..650

C. Ji, Enumerative Properties of Posets Corresponding to a Certain Class of No Strategy Games, arXiv:1608.06025 [math.CO], 2016

FORMULA

a(n) = f(n, n, 0) 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(n, n, 0):

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[n, n, 0];

Table[a[n], {n, 1, 35}] (* Jean-Fran├žois Alcover, Nov 10 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A276027, A276029.

Sequence in context: A102088 A297295 A088142 * A209902 A049370 A009343

Adjacent sequences:  A276025 A276026 A276027 * A276029 A276030 A276031

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 February 17 14:12 EST 2018. Contains 299296 sequences. (Running on oeis4.)