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

0,2

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.

References

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

Links

Table of n, a(n) for n=0..39.

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^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.

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A073815 A212994 A307640 * A195998 A291167 A361592

Adjacent sequences: A339628 A339629 A339630 * A339632 A339633 A339634

Keyword

nonn

Author

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

Status

approved