A253026 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.
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, 21, 53, 85, 85, 53, 21, 7, 8, 25, 69, 149, 85, 149, 69, 25, 8, 9, 29, 85, 213, 341, 341, 213, 85, 29, 9, 10, 33, 101, 277, 597, 341, 597, 277, 101, 33, 10, 11, 37, 117, 341, 853, 1365
Offset: 1
Examples
Table starts: .0..1...2...3....4....5.....6.....7.....8.....9.....10.....11.....12......13 .1..1...5...9...13...17....21....25....29....33.....37.....41.....45......49 .2..5...5..21...37...53....69....85...101...117....133....149....165.....181 .3..9..21..21...85..149...213...277...341...405....469....533....597.....661 .4.13..37..85...85..341...597...853..1109..1365...1621...1877...2133....2389 .5.17..53.149..341..341..1365..2389..3413..4437...5461...6485...7509....8533 .6.21..69.213..597.1365..1365..5461..9557.13653..17749..21845..25941...30037 .7.25..85.277..853.2389..5461..5461.21845.38229..54613..70997..87381..103765 .8.29.101.341.1109.3413..9557.21845.21845.87381.152917.218453.283989..349525 .9.33.117.405.1365.4437.13653.38229.87381.87381.349525.611669.873813.1135957 Some solutions for n=4 and k=4: ..0..1..2..2....0..1..1..2....0..0..1..2....0..1..2..2....0..1..1..2 ..1..1..2..2....0..1..1..2....0..0..1..2....1..1..2..2....0..1..2..2 ..2..2..2..2....1..1..1..2....1..1..1..2....1..2..2..2....1..1..2..2 ..2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2
Links
- R. H. Hardin, Table of n, a(n) for n = 1..10018
- Aaron Barnoff, Curtis Bright, and Jeffrey Shallit, Using finite automata to compute the base-b representation of the golden ratio and other quadratic irrationals, arXiv:2405.02727 [cs.FL], 2024. See p. 8.
- Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 21.
- Robert Dougherty-Bliss and Manuel Kauers, Hardinian Arrays, arXiv:2309.00487 [math.CO], 2023, Hardinian Arrays, El. J. Combinat. 31 (2) (2024) #P2.9
Formula
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
T(n,k) = T(k,n) for all n,k.