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A339634 Number of final Tic-Tac-Toe positions on a (2*n+1) X 3 board that result in a tie. 3
3, 16, 30, 72, 178, 444, 1114, 2808, 7098, 17984, 45656, 116106, 295718, 754226, 1926060, 4924188, 12602416, 32284214, 82777240, 212415744, 545495716, 1401849594, 3604921774, 9275890122, 23881602058, 61518226734, 158548607640, 408814563524, 1054590179342 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The number of (2*n+1) X 3 0,1-matrices with 3*n+2 1's and 3*n+1 0's and no consecutive horizontal, vertical, nor diagonal triples of 111 or 000.

LINKS

Table of n, a(n) for n=0..28.

Doron Zeilberger, Class Projects for Combinatorics Fall 2020 (Rutgers University); Local copy [Pdf file only, no active links]

Doron Zeilberger, Initial Maple Package for Project 5 Combinatorics Fall 2020 (Rutgers University); Local copy

Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University); Local copy [Pdf file only, no active links]

FORMULA

a(n) = [x^(2*n+1)*t^ceiling(3*(2*n+1)/2)] (4*t^17*x^11 + 4*t^16*x^11 + 8*t^16*x^10 + 12*t^15*x^10 + 6*t^15*x^9 + 8*t^14*x^10 + 8*t^14*x^9 + 8*t^13*x^9 - 2*t^13*x^8 + 6*t^12*x^9 - 16*t^12*x^8 - 2*t^11*x^8 - 26*t^11*x^7 - 26*t^10*x^7 - 19*t^10*x^6 - 38*t^9*x^6 - 7*t^9*x^5 - 19*t^8*x^6 - 13*t^8*x^5 - 13*t^7*x^5 - t^7*x^4 - 7*t^6*x^5 + 10*t^6*x^4 - t^5*x^4 + 16*t^5*x^3 + 16*t^4*x^3 + 9*t^4*x^2 + 18*t^3*x^2 + 9*t^2*x^2) / (t^12*x^8 + t^11*x^7 + t^10*x^7 + t^9*x^6 - 2*t^6*x^4 - t^5*x^3 - t^4*x^3 - t^3*x^2 + 1) for n >= 1.

EXAMPLE

For n = 0 it is a 3 X 1 matrix, and there are 3 arrangements of 2 1's and a single 0 such that there are no 3-streaks of 1's nor 0's in the matrix.

For n = 1 it is the classic 3 X 3 Tic-Tac-Toe board, having 1's as X's and 0's as O's.

MAPLE

# Maple program adapted from OddTTT3(N) in Project 5 of Doron Zeilberger's Combinatorics Class Fall 2020 (Rutgers University).

A339634List:=proc(n) local f, i, t, x, N:

f:=(4*t^17*x^11 + 4*t^16*x^11 + 8*t^16*x^10 + 12*t^15*x^10 + 6*t^15*x^9 + 8*t^14*x^10 + 8*t^14*x^9 + 8*t^13*x^9 - 2*t^13*x^8 + 6*t^12*x^9 - 16*t^12*x^8 - 2*t^11*x^8 - 26*t^11*x^7 - 26*t^10*x^7 - 19*t^10*x^6 - 38*t^9*x^6 - 7*t^9*x^5 - 19*t^8*x^6 - 13*t^8*x^5 - 13*t^7*x^5 - t^7*x^4 - 7*t^6*x^5 + 10*t^6*x^4 - t^5*x^4 + 16*t^5*x^3 + 16*t^4*x^3 + 9*t^4*x^2 + 18*t^3*x^2 + 9*t^2*x^2) / (t^12*x^8 + t^11*x^7 + t^10*x^7 + t^9*x^6 - 2*t^6*x^4 - t^5*x^3 - t^4*x^3 - t^3*x^2 + 1):

N:=n-1:

#Take the Taylor expansion up to x^(2*N+2)

f:=taylor(f, x=0, 2*N+3):

#Extract the coefficients of x^(2*i+1)*t^(3*i+2)

[3, seq(coeff(coeff(f, x, 2*i+1), t, 3*i+2), i=1..N)]:

end:

CROSSREFS

Bisection of A339631 (odd part).

Cf. A339633 (even part).

Sequence in context: A258717 A013196 A329866 * A196264 A031080 A013199

Adjacent sequences:  A339631 A339632 A339633 * A339635 A339636 A339637

KEYWORD

nonn

AUTHOR

Doron Zeilberger, Taerim Kim, Karnaa Mistry, Weiji Zheng, Dec 10 2020

STATUS

approved

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Last modified May 11 01:05 EDT 2021. Contains 343784 sequences. (Running on oeis4.)