cp's OEIS Frontend

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.

%I A222763 #32 Nov 29 2024 18:59:38
%S A222763 1,0,3,4,20,48,175,512,1719,5400,17776,57420,188656,617176,2033175,
%T A222763 6697744,22139780,73262232,242931321,806516560,2681475048,8925158440,
%U A222763 29740390672,99196158144,331163178475,1106489052968,3699881730900,12380449027324,41454579098852
%N 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.
%C A222763 From _Gus Wiseman_, Oct 05 2022: (Start)
%C A222763 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:
%C A222763   (135)    (234)    (333)    (432)    (531)
%C A222763   (11115)  (21114)  (31113)  (41112)  (51111)
%C A222763   (11214)  (21213)  (31212)  (41211)
%C A222763   (11313)  (21312)  (31311)
%C A222763   (11412)  (21411)
%C A222763   (11511)
%C A222763 This is the odd-indexed version of A357182, and the corresponding unordered count (partitions) is A357488.
%C A222763 (End)
%H A222763 Alois P. Heinz, <a href="/A222763/b222763.txt">Table of n, a(n) for n = 0..1881</a> (terms n = 1..210 from R. H. Hardin)
%F A222763 a(n) = A105422(2n,n). - _Alois P. Heinz_, Sep 24 2024
%F A222763 a(n) = (n+1)*A046736(n+2). - _Mark van Hoeij_, Nov 29 2024
%e A222763 All solutions for n=3
%e A222763 ..0..1....0..0....0..0....0..0
%e A222763 ..0..0....0..0....0..1....1..0
%e A222763 ..0..0....1..0....0..0....0..0
%p A222763 a:= proc(n) option remember; `if`(n<3, [1, 0, 3][n+1],
%p A222763       (4*(n-1)*(74*n^2-153*n+73)*a(n-1) +8*(2*n-3)*
%p A222763       (74*n^2-153*n+70)*a(n-2) -2*(37*n-21)*(2*n-5)*
%p A222763       (n-1)*a(n-3))/(5*(37*n-58)*n*(n-1)))
%p A222763     end:
%p A222763 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 24 2024
%Y A222763 Column k = 2 of A222769.
%Y A222763 Cf. A046736, A105422.
%K A222763 nonn
%O A222763 0,3
%A A222763 _R. H. Hardin_, Mar 05 2013
%E A222763 a(0)=1 prepended by _Alois P. Heinz_, Sep 24 2024