A339633 Number of final Tic-Tac-Toe positions on a (2*n) X 3 board that resulted in a tie.

1, 18, 28, 58, 140, 334, 824, 2038, 5084, 12730, 32004, 80694, 204004, 516902, 1312336, 3337682, 8502132, 21688182, 55395140, 141651742, 362601356, 929084578, 2382677360, 6115461118, 15707982020, 40375223374, 103846409504, 267258086338, 688201711116, 1773088924494

Offset

0,2

Comments

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

References

Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University).

Links

Table of n, a(n) for n=0..29.

Doron Zeilberger, Class Projects for Combinatorics Fall 2020 (Rutgers University).

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

Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University).

Formula

a(n) = [x^(2*n)*t^(3*n)] (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 = 1 it is a 3 X 2 matrix, and so there are 18 ways to have three 1's and three 0's such that there are no 3-streaks of 1's nor 0's in the matrix.

Maple

# Maple program adapted from EvenTTT3(N) in Project 5 of Doron Zeilberger's Combinatorics Class Fall 2020 (Rutgers University).
A339633List:=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)*t^(3*i)
[1, seq(coeff(coeff(f, x, 2*i), t, 3*i), i=1..N)]:
end:

Crossrefs

Bisection of A339631 (even part).

Cf. A339634 (odd part).

Sequence in context: A117101 A063840 A180117 * A167333 A259642 A045000

Adjacent sequences: A339630 A339631 A339632 * A339634 A339635 A339636

Keyword

nonn

Author

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

Status

approved