The Pierce expansion, or alternated Egyptian product, of a real number is the unique increasing sequence 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
(3)
|
where
(4)
| |||
(5)
|
and (Shallit 1984). This recurrence has explicit solution
(6)
|
not noted by Shallit (1984).
, corresponding to , has the particularly beautiful form
(7)
| |||
(8)
|
where is a Fibonacci number.
The following table gives coefficients and for some small integer .
OEIS | OEIS | ||||
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, ... |