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%I #6 Sep 04 2022 13:18:37
%S 1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,3,2,2,3,1,1,5,3,3,3,5,1,1,8,5,4,4,5,
%T 8,1,1,13,8,6,7,6,8,13,1,1,21,13,9,9,9,9,13,21,1,1,34,21,14,15,14,15,
%U 14,21,34,1,1,55,34,22,26,24,24,26,22,34,55,1,1,89,55,35,46,44,40,44,46,35,55
%N T(n,k) = Number of n X k 0..1 arrays with every element equal to 0, 3, 5, 7 or 8 king-move adjacent elements, with upper left element zero.
%C Table starts
%C .1..1..1..1..1..1...1...1...1....1....1.....1.....1......1......1.......1
%C .1..1..1..2..3..5...8..13..21...34...55....89...144....233....377.....610
%C .1..1..1..2..3..5...8..13..21...34...55....89...144....233....377.....610
%C .1..2..2..3..4..6...9..14..22...35...56....90...145....234....378.....611
%C .1..3..3..4..7..9..15..26..46...84..151...276...506....929...1708....3138
%C .1..5..5..6..9.14..24..44..81..156..306...602..1192...2370...4720....9415
%C .1..8..8..9.15.24..40..76.141..277..570..1171..2441...5157..10913...23193
%C .1.13.13.14.26.44..76.168.359..792.1895..4521.10886..26818..66131..163463
%C .1.21.21.22.46.81.141.359.873.2145.5971.16568.45898.131372.376833.1078872
%H R. H. Hardin, <a href="/A298917/b298917.txt">Table of n, a(n) for n = 1..612</a>
%F Empirical for column k:
%F k=1: a(n) = a(n-1).
%F k=2: a(n) = a(n-1) +a(n-2) for n>3.
%F k=3: a(n) = a(n-1) +a(n-2) for n>3.
%F k=4: a(n) = 2*a(n-1) -a(n-3).
%F k=5: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-5) -a(n-6) -a(n-7) -a(n-8) for n>9.
%F k=6: a(n) = 3*a(n-1) -2*a(n-2) +a(n-3) -2*a(n-4) -a(n-7) +2*a(n-8) for n>9.
%F k=7: [order 14] for n>15.
%e All solutions for n=5, k=4
%e ..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0
%e ..0..0..0..0. .0..0..1..1. .0..0..0..0. .0..0..0..0
%e ..1..1..1..1. .0..0..1..1. .0..0..0..0. .0..0..0..0
%e ..1..1..1..1. .0..0..1..1. .1..1..1..1. .0..0..0..0
%e ..1..1..1..1. .0..0..1..1. .1..1..1..1. .0..0..0..0
%Y Columns 2 and 3 are A000045(n-1).
%Y Column 4 is A001611(n-1).
%K nonn,tabl
%O 1,12
%A _R. H. Hardin_, Jan 29 2018
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