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A276027 Number of ways to transform a sequence of n ones to a single number by continually removing two numbers and replacing them with their sum modulo 3. 3
1, 1, 1, 2, 4, 7, 18, 43, 93, 266, 702, 1687, 5136, 14405, 36898, 117016, 341842, 914064, 2983027, 8972121, 24743851, 82478973, 253555061, 715745648, 2424954125, 7582390623, 21796481477, 74805170349, 237095926682, 691568408221, 2398418942361, 7686495623620 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Can be considered as the number of maximal chains in a poset whose nodes are the possible states of the sequence.  In this sense it counts the same things as A002846 when the elements of that poset are taken modulo 3.

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..1000

C. Ji, Example for a(7)

FORMULA

a(n) = f(0, 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.

EXAMPLE

For n = 4, the two ways are 1111 -> 211 -> 10 -> 1 and 1111 -> 211 -> 22 -> 1.

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, 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[0, n, 0]; Array[a, 35] (* Jean-Fran├žois Alcover, Aug 07 2017, after Alois P. Heinz *)

PROG

(Python)

from sympy.core.cache import cacheit

@cacheit

def b(x, y, z): return 1 if x + y + z==1 else (b(x + 1, y - 1, z - 1) if y>0 and z>0 else 0) + (b(x - 1, y, z) if x>1 or x>0 and y>0 or x>0 and z>0 else 0) + (b(x, y - 2, z + 1) if y>1 else 0) + (b(x, y + 1, z - 2) if z>1 else 0)

def a(n): return b(0, n, 0)

print([a(n) for n in range(1, 36)]) # Indranil Ghosh, Aug 09 2017, after Maple code

CROSSREFS

Cf. A002846, A117143.

Similar to A002846 with nodes taken modulo 3.

A117143 is the total number of nodes in this poset.

Sequence in context: A300913 A232484 A223013 * A101569 A225435 A243049

Adjacent sequences:  A276024 A276025 A276026 * A276028 A276029 A276030

KEYWORD

nonn

AUTHOR

Caleb Ji, Aug 16 2016

EXTENSIONS

a(19)-a(32) from Alois P. Heinz, Aug 18 2016

STATUS

approved

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Last modified May 18 19:02 EDT 2021. Contains 344001 sequences. (Running on oeis4.)