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.
1, 3, 10, 50, 259, 1540, 9594, 62649, 422598, 2960716, 21030711, 152470357, 1129502128, 8434189996, 63674017174, 488573782216, 3762932025753, 29178861276815, 229208503750838, 1803350026315019, 14248236439629534, 113825380196996557, 909507867095014380
Offset: 1
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..650
Programs
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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 *)
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.
Extensions
More terms from Alois P. Heinz, Aug 18 2016
Comments