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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. 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
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.
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
KEYWORD
nonn,tabl,changed
AUTHOR
R. H. Hardin, Dec 26 2014
STATUS
approved

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Last modified May 23 04:41 EDT 2024. Contains 372758 sequences. (Running on oeis4.)