A166310 Wythoff Triangle, T.

1, 2, 3, 4, 6, 8, 5, 7, 9, 11, 10, 12, 14, 16, 21, 13, 15, 17, 19, 24, 29, 18, 20, 22, 27, 32, 37, 42, 23, 25, 30, 35, 40, 45, 50, 55, 26, 28, 33, 38, 43, 48, 53, 58, 63, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 34, 39, 44, 49, 54, 59, 64, 69, 74, 79, 84, 47, 52, 57, 62, 67, 72

Offset

1,2

Comments

(1) Every positive integer occurs exactly once, so that

this is a permutation of the natural numbers.

(2) Obtained from the preliminary Wyhoff triangle

(A166309) by arranging each row in increasing order.

(3) The difference between consecutive row terms is a

Fibonacci number (A000045).

(4) Is the difference between consecutive column terms a

Fibonacci number?

References

C. 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.

Links

Table of n, a(n) for n=1..72.

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). Then for each a, arrange the numbers P

(a,b) in increasing order.

Example

The first nine rows of T:
1
2....3
4....6...8
5....7...9..11
10..12..14..16..21
13..15..17..19..24..29
18..20..22..27..32..37..42
23..25..30..35..40..45..50..55
26..28..33..38..43..48..53..58..63
Row 5 of the preliminary Wythoff triangle is
16,21,10,12,14, so that row 5 of the Wythoff triangle is
10,12,14,16,21. These are the row numbers of the Wythoff
array W (A035513) which precurse to pairs (5,b) for
b=0,1,2,3,4, not respectively. Example of precursion: row
16 of W is 40,65,105,...; then 65-40=25, 40-25=15,
25-15=10, 15-10=5, 10-5=5, 5-5=0, 5-0=5, so that the
initial pair (5,0) is reached in seven precursive steps.

Mathematica

f[n_]:=f[n]=Fibonacci[n]; w[n_, k_] := f[k + 1] Floor[n GoldenRatio] + (n - 1) f[k]; a[n_, k_] := w[n, Module[{z = 0}, ((While[w[#1, z] <= w[#1, z + 1], z--]; z - 1) &)[n] + k]]; z = 100; t = Table[a[n, k], {n, 1, z}, {k, 1, 2}] (* n-th pair: 1st 2 terms of row n of left-justified Wythoff  array, A165357 *)
u = Table[t[[n]][[1]], {n, 1, z}]
v = Table[Flatten[Position[u, n]], {n, 1, z/5}]
Flatten[v]  (* A166310 sequence *)
TableForm[Table[Flatten[Position[u, n]], {n, 1, z/5}]]  (* A166310 triangle, Clark Kimberling, Aug 01 2013 *)

Crossrefs

Cf. A035513, A165357, A166309.

Sequence in context: A120817 A327887 A339361 * A293030 A109852 A083197

Adjacent sequences: A166307 A166308 A166309 * A166311 A166312 A166313

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 11 2009

Status

approved