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A250742 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nonincreasing x(i,j)-x(i-1,j) in the j direction.

%I A250742 #6 Jul 23 2025 12:45:07
%S A250742 6,10,10,18,14,18,34,22,22,34,66,38,30,38,66,130,70,46,46,70,130,258,
%T A250742 134,78,62,78,134,258,514,262,142,94,94,142,262,514,1026,518,270,158,
%U A250742 126,158,270,518,1026,2050,1030,526,286,190,190,286,526,1030,2050,4098,2054,1038
%N A250742 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nonincreasing x(i,j)-x(i-1,j) in the j direction.
%C A250742 Table starts
%C A250742 ....6...10...18...34...66..130..258..514.1026.2050.4098..8194.16386.32770.65538
%C A250742 ...10...14...22...38...70..134..262..518.1030.2054.4102..8198.16390.32774.65542
%C A250742 ...18...22...30...46...78..142..270..526.1038.2062.4110..8206.16398.32782.65550
%C A250742 ...34...38...46...62...94..158..286..542.1054.2078.4126..8222.16414.32798.65566
%C A250742 ...66...70...78...94..126..190..318..574.1086.2110.4158..8254.16446.32830.65598
%C A250742 ..130..134..142..158..190..254..382..638.1150.2174.4222..8318.16510.32894.65662
%C A250742 ..258..262..270..286..318..382..510..766.1278.2302.4350..8446.16638.33022.65790
%C A250742 ..514..518..526..542..574..638..766.1022.1534.2558.4606..8702.16894.33278.66046
%C A250742 .1026.1030.1038.1054.1086.1150.1278.1534.2046.3070.5118..9214.17406.33790.66558
%C A250742 .2050.2054.2062.2078.2110.2174.2302.2558.3070.4094.6142.10238.18430.34814.67582
%H A250742 R. H. Hardin, <a href="/A250742/b250742.txt">Table of n, a(n) for n = 1..1450</a>
%F A250742 The constraints apparently result in horizontally or vertically banded arrays, hence:
%F A250742 Empirical: T(n,k) = 2^(k+1)+2^(n+1)-2
%F A250742 Empirical for column k:
%F A250742 k=1: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +2
%F A250742 k=2: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +6
%F A250742 k=3: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +14
%F A250742 k=4: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +30
%F A250742 k=5: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +62
%F A250742 k=6: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +126
%F A250742 k=7: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +254
%e A250742 Some solutions for n=4 k=4
%e A250742 ..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1
%e A250742 ..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1
%e A250742 ..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....0..0..0..0..0
%e A250742 ..1..1..1..1..1....1..0..1..0..1....0..1..0..0..1....0..0..0..0..0
%e A250742 ..1..1..1..1..1....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1
%Y A250742 Column 1 is A052548(n+1)
%Y A250742 Column 2 is A153972(n+1)
%Y A250742 Diagonal is A000918(n+2)
%K A250742 nonn,tabl
%O A250742 1,1
%A A250742 _R. H. Hardin_, Nov 27 2014