A339631 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.

1, 3, 18, 16, 28, 30, 58, 72, 140, 178, 334, 444, 824, 1114, 2038, 2808, 5084, 7098, 12730, 17984, 32004, 45656, 80694, 116106, 204004, 295718, 516902, 754226, 1312336, 1926060, 3337682, 4924188, 8502132, 12602416, 21688182, 32284214, 55395140, 82777240, 141651742, 212415744

Offset: 0

Author

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

Keywords

• nonn

Comments

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.

Examples

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

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, Initial Maple Package for Project 5 Combinatorics Fall 2020 (Rutgers University).
• Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University).

Crossrefs

Bisections give: A339633 (even part), A339634 (odd part).

Programs

Julia

using Nemo
function A339631List(prec)
    R, t = PolynomialRing(ZZ, "t")
    S, x = PowerSeriesRing(R, prec+1, "x")
    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)
    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)
    ser = divexact(num, den)
    C = [coeff(coeff(ser, n), div(3*n, 2)) for n in 0:prec]
    C[1] = 1; C[2] = 3
    return C
end
A339631List(39) |> println # Peter Luschny, Dec 19 2020

Formula

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.