A120873 Fractal sequence of the Wythoff difference array (A080164).

1, 1, 2, 3, 1, 4, 2, 5, 6, 3, 7, 8, 1, 9, 4, 10, 11, 2, 12, 5, 13, 14, 6, 15, 16, 3, 17, 7, 18, 19, 8, 20, 21, 1, 22, 9, 23, 24, 4, 25, 10, 26, 27, 11, 28, 29, 2, 30, 12, 31, 32, 5, 33, 13, 34, 35, 14, 36, 37, 6, 38, 15, 39, 40, 16, 41, 42, 3, 43, 17, 44, 45, 7, 46, 18, 47, 48, 19, 49

Offset: 1

Author

Clark Kimberling, Jul 10 2006

Keywords

• nonn

Comments

A fractal sequence f contains itself as a proper subsequence; e.g., if you delete the first occurrence of each positive integer, the remaining sequence is f; thus f properly contains itself infinitely many times.

This is subsequence of A003603. - Clark Kimberling, Oct 26 2021

a(n) is the number of the row of the Wythoff array (A035513) that contains the n-th Wythoff pair; e.g., the 6th Wythoff pair is (floor(6*r), floor(6*r^2)), where r = golden ratio = A001622, which is in row 4 of the Wythoff array. - Clark Kimberling, Oct 26 2021

Examples

			The fractal sequence f(n) of a dispersion D={d(g,h,)} is defined as follows.
For each positive integer n there is a unique (g,h) such that n=d(g,h) and f(n)=g. So f(7)=2 because the row of the WDA in which 7 occurs is row 2.
		

References

• Clark Kimberling, The Wythoff difference array, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 153-158.

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Classic Sequences.

Crossrefs

Cf. A001622, A003603, A035513, A080164.

Programs

PARI

lowerw(n) = (n+sqrtint(5*n^2))\2 ; \\ A000201
upperw(n) = (sqrtint(n^2*5)+n*3)\2; \\ A001950
compoundw(n) = (sqrtint(n^2*5)+n*3)\2 - 1; \\ A003622
a(n) = my(x=lowerw(n), y=upperw(n), u); while (1, my(k=1, ok=1); while(ok, my(xx = lowerw(k), yy = compoundw(k)); if ((x == xx) && (y == yy), return(k)); if (xx > x, ok = 0); k++;); u = x; x = y - u; y = u;); \\ Michel Marcus, Sep 17 2022