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A120873
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Fractal sequence of the Wythoff difference array (A080164).
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2
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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
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OFFSET
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1,3
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COMMENTS
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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.
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
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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