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

1,4

Links

R. H. Hardin, Table of n, a(n) for n = 1..10018

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.

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.

Example

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

Crossrefs

Column 1 is A000027(n-1).

Column 2 is A004766(n-2).

Diagonal is A002450(n-1).

Sequence in context: A075758 A125596 A351962 * A341145 A204994 A277253

Adjacent sequences: A253023 A253024 A253025 * A253027 A253028 A253029

Keyword

nonn,tabl

Author

R. H. Hardin, Dec 26 2014

Status

approved