A222763 Number of n X 2 0..1 arrays with exactly floor(nX2/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

1, 0, 3, 4, 20, 48, 175, 512, 1719, 5400, 17776, 57420, 188656, 617176, 2033175, 6697744, 22139780, 73262232, 242931321, 806516560, 2681475048, 8925158440, 29740390672, 99196158144, 331163178475, 1106489052968, 3699881730900, 12380449027324, 41454579098852

Offset: 0

Author

R. H. Hardin, Mar 05 2013

Keywords

• nonn

Comments

From Gus Wiseman, Oct 05 2022: (Start)

Conjecture: Also the number of integer compositions of 2n + 1 with the same length as reverse-alternating sum. Here, the reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i. For example, the a(4) = 20 compositions are:

(135) (234) (333) (432) (531)

(11115) (21114) (31113) (41112) (51111)

(11214) (21213) (31212) (41211)

(11313) (21312) (31311)

(11412) (21411)

(11511)

This is the odd-indexed version of A357182, and the corresponding unordered count (partitions) is A357488.

(End)

Examples

			All solutions for n=3
..0..1....0..0....0..0....0..0
..0..0....0..0....0..1....1..0
..0..0....1..0....0..0....0..0
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1881 (terms n = 1..210 from R. H. Hardin)

Crossrefs

Column k = 2 of A222769.

Cf. A046736, A105422.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, [1, 0, 3][n+1],
      (4*(n-1)*(74*n^2-153*n+73)*a(n-1) +8*(2*n-3)*
      (74*n^2-153*n+70)*a(n-2) -2*(37*n-21)*(2*n-5)*
      (n-1)*a(n-3))/(5*(37*n-58)*n*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 24 2024

Formula

a(n) = A105422(2n,n). - Alois P. Heinz, Sep 24 2024

a(n) = (n+1)*A046736(n+2). - Mark van Hoeij, Nov 29 2024

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 24 2024