

A253026


T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right its kingmove 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.


3



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
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OFFSET

1,4


LINKS



FORMULA

T(n,k) = (nk)*4^(k1) + (4^(k1)1)/3 for all n>=k>=1 (Thm. 2 in the paper of DougertyBliss 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



KEYWORD



AUTHOR



STATUS

approved



