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T(n,k) = half the number of (n+1) X (k+1) binary arrays with no 2 X 2 subblock having exactly 2 ones.
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%I #14 Jan 09 2025 12:32:56

%S 5,13,13,33,47,33,85,161,161,85,217,567,730,567,217,557,1969,3435,

%T 3435,1969,557,1425,6887,15887,21935,15887,6887,1425,3653,24001,74148,

%U 136843,136843,74148,24001,3653,9353,83799,344483,864671,1146964,864671,344483

%N T(n,k) = half the number of (n+1) X (k+1) binary arrays with no 2 X 2 subblock having exactly 2 ones.

%C All columns (or rows) are linear recurrences with constant coefficients with the order of the recurrence <= A005418(k+1). For columns up to k=6, the only case where the order is strictly less than this upper bound is for column 4. - _Andrew Howroyd_, Jan 09 2025

%H R. H. Hardin, <a href="/A183782/b183782.txt">Table of n, a(n) for n = 1..364</a>

%e Table starts:

%e 5 13 33 85 217 557 1425

%e 13 47 161 567 1969 6887 24001

%e 33 161 730 3435 15887 74148 344483

%e 85 567 3435 21935 136843 864671 5431499

%e 217 1969 15887 136843 1146964 9764363 82573675

%e 557 6887 74148 864671 9764363 112439612 1284649009

%e 1425 24001 344483 5431499 82573675 1284649009 19815396763

%e 3653 83799 1604473 34228999 701022093 14750484447 307446383061

%e 9353 292305 7462786 215374371 5941108591 169027656900 4760294305144

%e 23965 1020103 34738575 1356329167 50402814448 1939448495087 73819487702999

%e ...

%e Some solutions with a(1,1)=0 for 4 X 3:

%e ..0..0..1....0..0..1....0..0..0....0..0..0....0..0..1....0..0..1....0..1..1

%e ..1..0..0....0..0..0....0..1..0....0..0..0....0..1..1....0..0..0....1..1..1

%e ..0..0..1....1..0..1....1..1..1....1..0..1....1..1..0....0..0..0....0..1..1

%e ..0..0..0....0..0..0....1..1..1....1..1..1....1..0..0....0..1..0....0..0..1

%Y Main diagonal is A183773.

%Y Columns 1..8 are A183774, A183775, A183776, A183777, A183778, A183779, A183780, A183781.

%Y Cf. A005418.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Jan 07 2011