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Pierce Expansion

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The Pierce expansion, or alternated Egyptian product, of a real number 0<x<1 is the strictly increasing sequence {a_1,a_2,...} of positive integers a_i obtained by the greedy algorithm x_0=x, a_n=|_1/x_(n-1)_|, and x_n=1-a_nx_(n-1), stopping when x_n=0, where |_x_| is the floor function. It gives the representation

x=1/(a_1)-1/(a_1a_2)+1/(a_1a_2a_3)-....
(1)

This convention makes the expansion unique; for example, it assigns {2} to x=1/2, although the strictly increasing sequence {1,2} gives the same value in the displayed sum. A real number 0<x<1 has a finite Pierce expansion iff x is rational.

Special cases are summarized in the following table, where K is Catalan's constant, gamma is the Euler-Mascheroni constant, and phi is the golden ratio.

xOEISPierce expansion
2^(-1/2)A0918311, 3, 8, 33, 35, 39201, 39203, 60245508192801, ...
Catalan's constant KA1322011, 11, 13, 59, 582, 12285, 127893, 654577, ...
cos1A1182391, 2, 12, 30, 56, 90, 132, 182, 240, ...
e^(-1)A0207252, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
Euler-Mascheroni constant gammaA0062841, 2, 6, 13, 21, 24, 225, 615, 17450, ...
natural logarithm of 2 ln2A0918461, 3, 12, 21, 51, 57, 73, 85, 96, ...
phi^(-1)A1182421, 2, 4, 17, 19, 5777, 5779, 192900153617, ...
pi^(-1)A0062833, 22, 118, 383, 571, 635, 70529, ...
sech11, 2, 3, 8, 9, 24, 37, 85, ...
sin1A0683771, 6, 20, 42, 72, 110, 156, 210, 272, ...

If c>=3 is an integer and x is of the form

x=(c-sqrt(c^2-4))/2,
(2)

then there is a closed form for the Pierce expansion given by

{a_n}_(n=1)^infty={c_0-1,c_0+1,c_1-1,c_1+1,c_2-1,c_2+1,...},
(3)

where c_0=c=(1+x^2)/x and c_(k+1)=c_k^3-3c_k for nonnegative integers k (Shallit 1984). This recurrence relation has explicit solution

c_k^((c))=-2cos[3^kcos^(-1)(-1/2c)]
(4)

not noted by Shallit (1984).

c=3, corresponding to x=(3-sqrt(5))/2, has the particularly beautiful form

c_n^((3))=-2cos[3^ncos^(-1)(-3/2)]
(5)
=2F_(2·3^n+1)-F_(2·3^n),
(6)

where F_n is a Fibonacci number.

The following table gives coefficients c_k and a_k for some small integer c.

cxOEIS{c_k}OEIS{a_k}
31/2(3-sqrt(5))A0019993, 18, 5778, 192900153618, ...A0062762, 4, 17, 19, 5777, 5779, ...
42-sqrt(3)4, 52, 140452, 2770663499604052, ...3, 5, 51, 53, 140451, 140453, ...
51/2(5-sqrt(21))5, 110, 1330670, 2356194280407770990, ...4, 6, 109, 111, 1330669, 1330671, ...
63-2sqrt(2)A1128456, 198, 7761798, 467613464999866416198, ...A0062755, 7, 197, 199, 7761797, 7761799, ...

For x=3-2sqrt(2), the coefficients a_k are the terms of OEIS A006275 after omitting its initial 2.

See also

Engel Expansion

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References

Erdős, P. and Shallit, J. O. "New Bounds on the Length of Finite Pierce and Engel Series." Sem. Theor. Nombres Bordeaux 3, 43-53, 1991.Keselj, V. "Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations." Sep. 10, 1996. https://cs.uwaterloo.ca/research/tr/1996/21/cs-96-21.pdf.Mays, M. E. "Iterating the Division Algorithm." Fib. Quart. 25, 204-213, 1987.Pierce, T. A. "On an Algorithm and Its Use in Approximating Roots of Algebraic Equations." Amer. Math. Monthly 36, 523-525, 1929. https://doi.org/10.1080/00029890.1929.11987017.Salzer, H. E. "The Approximation of Numbers as Sums of Reciprocals." Amer. Math. Monthly 54, 135-142, 1947.Shallit, J. O. "Some Predictable Pierce Expansions." Fib. Quart. 22, 332-335, 1984.Shallit, J. O. "Metric Theory of Pierce Expansions." Fib. Quart. 24, 22-40, 1986.Sloane, N. J. A. Sequences A001999/M3055, A006275/M1342, A006276/M1298, A006283/M3092, A006284/M1593, A020725, A068377, A091831, A091846, A112845, A118239, A118242, and A132201 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Pierce Expansion

Cite this as:

Weisstein, Eric W. "Pierce Expansion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PierceExpansion.html

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