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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.
3
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
0,3
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)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1881 (terms n = 1..210 from R. H. Hardin)
FORMULA
a(n) = A105422(2n,n). - Alois P. Heinz, Sep 24 2024
EXAMPLE
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
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
CROSSREFS
Column k = 2 of A222769.
Cf. A105422.
Sequence in context: A262033 A062870 A226964 * A306668 A300499 A151419
KEYWORD
nonn
AUTHOR
R. H. Hardin, Mar 05 2013
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Sep 24 2024
STATUS
approved