Pierce Expansion
المؤلف:
Erdős, P. and Shallit, J. O.
المصدر:
"New Bounds on the Length of Finite Pierce and Engel Series." Sem. Theor. Nombres Bordeaux 3
الجزء والصفحة:
...
12-5-2020
981
Pierce Expansion
The Pierce expansion, or alternated Egyptian product, of a real number 
is the unique increasing sequence ![<span style=]()
{a_1,a_2,...}" src="https://mathworld.wolfram.com/images/equations/PierceExpansion/Inline2.gif" style="height:15px; width:66px" /> of positive integers 
such that
 |
(1)
|
A number 
has a finite Pierce expansion iff 
is rational.
Special cases are summarized in the following table.
 |
OEIS |
Pierce expansion |
 |
A091831 |
1, 3, 8, 33, 35, 39201, 39203, 60245508192801, ... |
Catalan's constant  |
A132201 |
1, 11, 13, 59, 582, 12285, 127893, 654577, ... |
 |
A118239 |
1, 2, 12, 30, 56, 90, 132, 182, 240, ... |
 |
A020725 |
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... |
Euler-Mascheroni constant  |
A006284 |
1, 2, 6, 13, 21, 24, 225, 615, 17450, ... |
natural logarithm of 2  |
A091846 |
1, 3, 12, 21, 51, 57, 73, 85, 96, ... |
 |
A118242 |
1, 2, 4, 17, 19, 5777, 5779, 192900153617, ... |
 |
A006283 |
3, 22, 118, 383, 571, 635, 70529, ... |
 |
|
1, 2, 3, 8, 9, 24, 37, 85, ... |
 |
A068377 |
1, 6, 20, 42, 72, 110, 156, 210, 272, ... |
If 
is of the form
 |
(2)
|
then there is a closed-form for the Pierce expansion given by
![x=<span style=]() {c_0-1,c_0+1,c_1-1,c_1+1,c_2-1,c_2+1,...}, " src="https://mathworld.wolfram.com/images/equations/PierceExpansion/NumberedEquation3.gif" style="height:15px; width:304px" /> |
(3)
|
where
and 
(Shallit 1984). This recurrence has explicit solution
![c_k^((c))=-2cos[3^kcos^(-1)(-1/2c)]](https://mathworld.wolfram.com/images/equations/PierceExpansion/NumberedEquation4.gif) |
(6)
|
not noted by Shallit (1984).
, corresponding to 
, has the particularly beautiful form
where 
is a Fibonacci number.
The following table gives coefficients 
and 
for some small integer 
.
 |
 |
OEIS |
![<span style=]() {c_k}" src="https://mathworld.wolfram.com/images/equations/PierceExpansion/Inline39.gif" style="height:15px; width:22px" /> |
OEIS |
![<span style=]() {a_k}" src="https://mathworld.wolfram.com/images/equations/PierceExpansion/Inline40.gif" style="height:15px; width:23px" /> |
| 3 |
 |
A001999 |
3, 18, 5778, 192900153618, ... |
A006276 |
2, 4, 17, 19, 5777, 5779, ... |
| 4 |
 |
|
4, 52, 140452, 2770663499604052, ... |
|
3, 5, 51, 53, 140451, 140453, ... |
| 5 |
 |
|
5, 110, 1330670, 2356194280407770990, ... |
|
4, 6, 109, 111, 1330669, 1330671, ... |
| 6 |
 |
A112845 |
6, 198, 7761798, 467613464999866416198, ... |
A006275 |
5, 5, 7, 197, 199, 7761797, ... |
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://www.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 Polynomials." Amer. Math. Monthly 36, 523-525, 1929.
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, A006283/M3092, A006284/M1593, A006276/M1298, A020725, A091831, A091846, A112845, A118242, and A132201 in "The On-Line Encyclopedia of Integer Sequences."
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