OFFSET
1,4
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.
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
nonn,tabl
AUTHOR
R. H. Hardin, Dec 26 2014
STATUS
approved