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%I #30 May 22 2024 11:42:39
%S 0,1,1,2,1,2,3,5,5,3,4,9,5,9,4,5,13,21,21,13,5,6,17,37,21,37,17,6,7,
%T 21,53,85,85,53,21,7,8,25,69,149,85,149,69,25,8,9,29,85,213,341,341,
%U 213,85,29,9,10,33,101,277,597,341,597,277,101,33,10,11,37,117,341,853,1365
%N T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 1 and every value within 1 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.
%H R. H. Hardin, <a href="/A253026/b253026.txt">Table of n, a(n) for n = 1..10018</a>
%H Aaron Barnoff, Curtis Bright, and Jeffrey Shallit, <a href="https://arxiv.org/abs/2405.02727">Using finite automata to compute the base-b representation of the golden ratio and other quadratic irrationals</a>, arXiv:2405.02727 [cs.FL], 2024. See p. 8.
%H Robert Dougherty-Bliss, <a href="https://sites.math.rutgers.edu/~zeilberg/Theses/RobertDoughertyBlissThesis.pdf">Experimental Methods in Number Theory and Combinatorics</a>, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 21.
%H Robert Dougherty-Bliss and Manuel Kauers, <a href="https://arxiv.org/abs/2309.00487">Hardinian Arrays</a>, arXiv:2309.00487 [math.CO], 2023, <a href="https://doi.org/10.37236/12358">Hardinian Arrays</a>, El. J. Combinat. 31 (2) (2024) #P2.9
%F T(n,k) = (n-k)*4^(k-1) + (4^(k-1)-1)/3 for all n>=k>=1 (Thm. 2 in the paper of Dougerty-Bliss and Kauers cited above). - _Manuel Kauers_, Sep 06 2023
%F T(n,k) = T(k,n) for all n,k.
%e Table starts:
%e .0..1...2...3....4....5.....6.....7.....8.....9.....10.....11.....12......13
%e .1..1...5...9...13...17....21....25....29....33.....37.....41.....45......49
%e .2..5...5..21...37...53....69....85...101...117....133....149....165.....181
%e .3..9..21..21...85..149...213...277...341...405....469....533....597.....661
%e .4.13..37..85...85..341...597...853..1109..1365...1621...1877...2133....2389
%e .5.17..53.149..341..341..1365..2389..3413..4437...5461...6485...7509....8533
%e .6.21..69.213..597.1365..1365..5461..9557.13653..17749..21845..25941...30037
%e .7.25..85.277..853.2389..5461..5461.21845.38229..54613..70997..87381..103765
%e .8.29.101.341.1109.3413..9557.21845.21845.87381.152917.218453.283989..349525
%e .9.33.117.405.1365.4437.13653.38229.87381.87381.349525.611669.873813.1135957
%e Some solutions for n=4 and k=4:
%e ..0..1..2..2....0..1..1..2....0..0..1..2....0..1..2..2....0..1..1..2
%e ..1..1..2..2....0..1..1..2....0..0..1..2....1..1..2..2....0..1..2..2
%e ..2..2..2..2....1..1..1..2....1..1..1..2....1..2..2..2....1..1..2..2
%e ..2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2
%Y Column 1 is A000027(n-1).
%Y Column 2 is A004766(n-2).
%Y Diagonal is A002450(n-1).
%K nonn,tabl
%O 1,4
%A _R. H. Hardin_, Dec 26 2014
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