A135511 Number of Pierce-Engel hybrid expansions of 3/b, b>=3.

1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3

Offset

3,2

Comments

Gives the number of representations of 3/b (for b>=3) as a sum of fractions 3/b= a_1/q_1 + a_2/(q_1 q_2) + a_3/(q_1 q_2 q_3) + ... a_n/(q_1 q_2 ... q_n), where each a_i is either 1 or -1 and the q_i are chosen greedily.

Equivalently, the q_i can be found by taking r_1 = 3 and applying either b=r_i q_i + r_(i+1) or b=r_i q_i - r(i-1), where 0<=r_(i-1)<r_i. (When the first equation is used to find q_i, then a_(i+1) will be of opposite sign than a_i. If the second is used, a_(i+1) will be of the same sign as a_i.) The process terminates when some r_(n+1)=0.

Links

Table of n, a(n) for n=3..107.

Eric Weisstein's World of Mathematics, Pierce Expansion.

Eric Weisstein's World of Mathematics, Engel Expansion.

Formula

h(n) = h(n mod 6), for n, (n mod 6) >= 3.

Example

5 = 3(1) + 2 -> 2(2) + 1 -> 1(5) + 0 or
5 = 3(1) + 2 -> 2(3) - 1 -> 1(5) + 0 or
5 = 3(2) - 1 -> 1(5) + 0
So 3/5 = 1 - 1/2 + 1/10, 1 - 1/3 - 1/15, 1/2 + 1/10: thus h(5)=3.

Crossrefs

Sequence in context: A057039 A369010 A260449 * A007413 A277750 A325520

Adjacent sequences: A135508 A135509 A135510 * A135512 A135513 A135514

Keyword

easy,nonn

Author

A. Sutyak (asutyak(AT)gmail.com), Feb 09 2008

Status

approved