cp's OEIS Frontend

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.

%I A276029 #24 Sep 07 2018 03:24:13
%S A276029 1,4,27,228,2226,23778,270693,3229106,39922172,507680620,6604676830,
%T A276029 87549425068,1178880306174,16086844260290,222045139578443,
%U A276029 3095457073064120,43529719213465854,616853383573066504,8801227720060618544,126344910516550743232
%N 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.
%C A276029 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
%H A276029 Alois P. Heinz, <a href="/A276029/b276029.txt">Table of n, a(n) for n = 1..800</a>
%F A276029 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.
%F A276029 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.
%p A276029 b:= proc(x, y, z) option remember;
%p A276029       `if`(x+y+z=1, 1, `if`(y>0 and z>0, b(x+1, y-1, z-1), 0)+
%p A276029       `if`(x>1 or x>0 and y>0 or x>0 and z>0, b(x-1, y, z), 0)+
%p A276029       `if`(y>1, b(x, y-2, z+1), 0)+`if`(z>1, b(x, y+1, z-2), 0))
%p A276029     end:
%p A276029 a:= n-> b(0, n, n):
%p A276029 seq(a(n), n=1..35);  # _Alois P. Heinz_, Aug 18 2016
%t A276029 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]];
%t A276029 a[n_] := b[0, n, n];
%t A276029 Table[a[n], {n, 1, 35}] (* _Jean-François Alcover_, Nov 10 2017, after _Alois P. Heinz_ *)
%Y A276029 Cf. A276027, A276028.
%K A276029 nonn
%O A276029 1,2
%A A276029 _Caleb Ji_, Aug 17 2016
%E A276029 More terms from _Alois P. Heinz_, Aug 18 2016