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%I #54 Dec 20 2020 02:36:41
%S 1,3,18,16,28,30,58,72,140,178,334,444,824,1114,2038,2808,5084,7098,
%T 12730,17984,32004,45656,80694,116106,204004,295718,516902,754226,
%U 1312336,1926060,3337682,4924188,8502132,12602416,21688182,32284214,55395140,82777240,141651742,212415744
%N Number of 3 X n matrices with 3*n/2 1's (if n is even) or (3*n+1)/2 1's (if n is odd) that do not have a horizontal nor vertical nor diagonal 3-streak of 1's.
%C Provided by D. Zeilberger's Maple package (ComboProject5.txt) for Combinatorics Fall 2020 at Rutgers University (see links). Generated using alternating procedures EvenTTT3() and OddTTT3() from this Maple package.
%D Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University).
%H Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/Combo20/ClassProjects.html">Class Projects for Combinatorics Fall 2020 (Rutgers University)</a>.
%H Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/Combo20/projects/ComboProject5.txt">Initial Maple Package for Project 5 Combinatorics Fall 2020 (Rutgers University)</a>.
%H Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/math454_20.html">Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University)</a>.
%F a(n) = [x^n*t^ceiling(3*n/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 >= 2.
%e For n = 1 it is a 3 X 1 matrix, and there is no way to have a 3-streak of 1's or 0's since there must be 2 1's and 1 0, so there are three matrices [110],[011],[101].
%e For n = 3 it is the classic Tic-Tac-Toe board, with 1's being X's and 0's being O's.
%o (Julia)
%o using Nemo
%o function A339631List(prec)
%o R, t = PolynomialRing(ZZ, "t")
%o S, x = PowerSeriesRing(R, prec+1, "x")
%o num = (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)
%o den = (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)
%o ser = divexact(num, den)
%o C = [coeff(coeff(ser, n), div(3*n, 2)) for n in 0:prec]
%o C[1] = 1; C[2] = 3
%o return C
%o end
%o A339631List(39) |> println # _Peter Luschny_, Dec 19 2020
%Y Bisections give: A339633 (even part), A339634 (odd part).
%K nonn
%O 0,2
%A _Doron Zeilberger_, Taerim Kim, _Karnaa Mistry_, Weiji Zheng, Dec 10 2020
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