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 A339633 Number of final Tic-Tac-Toe positions on a (2*n) X 3 board that resulted in a tie. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 Doron Zeilberger, Class Projects for 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

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Last modified May 7 13:08 EDT 2021. Contains 343650 sequences. (Running on oeis4.)