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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.
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
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-François Alcover, Nov 10 2017, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A102088 A297295 A088142 * A209902 A049370 A009343
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 June 13 17:32 EDT 2024. Contains 373391 sequences. (Running on oeis4.)