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A166309
Preliminary Wythoff Triangle, P.
1
1, 3, 2, 4, 6, 8, 5, 7, 9, 11, 16, 21, 10, 12, 14, 19, 24, 29, 13, 15, 17, 22, 27, 32, 37, 42, 18, 20, 25, 30, 35, 40, 45, 50, 55, 23, 28, 33, 38, 43, 48, 53, 58, 63, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 34, 39, 44, 49, 54, 59, 64, 69, 74, 79, 84, 97, 110, 47, 52, 57, 62
OFFSET
1,2
COMMENTS
Every positive integer occurs exactly once, so that this is a permutation of the natural numbers.
Arranging each row in increasing order results in the Wythoff triangle (A166310).
REFERENCES
Clark Kimberling, "The Wythoff triangle and unique representations of positive integers," Proceedings of the Fourteenth International Conference on Fibonacci Numbers and Their Applications," Aportaciones Matematicas Invertigacion 20 (2011) 155-169.
FORMULA
For a=1,2,3,... and b=0,1,...,a-1, let P(a,b) be the number of the row of the Wythoff array (A035513) that precurses to (a,b).
EXAMPLE
The first six rows of P:
1
3....2
4....6...8
5....7...9..11
16..21..10..12..14
19..24..29..13..15..17
The Wythoff array W begins with
1...2...3...5...
4...7..11..18...
6..10..16..26...
These rows precurse to rows of the left-justified Wythoff array (A165357):
1...0...1...1...1...2...3...
2...1...3...4...7..11..18...
2...0...2...2...4...6..10...
P(2,0)=3 because row 3 of W precurses to (2,0).
P(2,1)=2 because row 2 of W precurses to (2,1).
CROSSREFS
Sequence in context: A254052 A371173 A105746 * A349472 A195208 A114651
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Oct 11 2009
STATUS
approved