

A135511


Number of PierceEngel hybrid expansions of 3/b, b>=3.


0



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
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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(i1), where 0<=r_(i1)<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.
Weisstein, Eric W., Pierce Expansion.
Weisstein, Eric W., 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: A128222 A057039 A260449 * A007413 A277750 A072457
Adjacent sequences: A135508 A135509 A135510 * A135512 A135513 A135514


KEYWORD

easy,nonn


AUTHOR

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


STATUS

approved



