A339634 Number of final Tic-Tac-Toe positions on a (2*n+1) X 3 board that result in a tie.

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

Offset: 0

Author

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

Keywords

• nonn

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.

Examples

			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.
		

Links

• 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]

Crossrefs

Bisection of A339631 (odd part).

Cf. A339633 (even part).

Programs

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:

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.