A135513 Number of Pierce-Engel hybrid expansions of 4/b, b>=4.

1, 4, 2, 4, 1, 2, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 1

Offset

4,2

Comments

Gives the number of representations of 4/b (for b>=4) as a sum of fractions 4/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 = 4 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=4..108.

Eric Weisstein's World of Mathematics, Pierce Expansion.

Eric Weisstein's World of Mathematics, Engel Expansion.

Formula

h(n) = h(n mod 12), for n, (n mod 12) >= 4.

Example

13 = 4(3) + 1 -> 1(13) + 0 or
13 = 4(4) - 3 -> 3(4) + 1 -> 1(13) + 0 or
13 = 4(4) - 3 -> 3(5) - 2 -> 2(6) + 1 -> 1(13) + 0 or
13 = 4(4) - 3 -> 3(5) - 2 -> 2(7) - 1 -> 1(13) + 0
So 4/13 = 1/3 - 1/39, 1/4 + 1/16 - 1/208, 1/4 + 1/20 + 1/120 - 1/1560, 1/4 + 1/20 + 1/140 + 1/1820: thus h(13) = 4

Crossrefs

Sequence in context: A068454 A090976 A156199 * A176895 A335261 A351864

Adjacent sequences: A135510 A135511 A135512 * A135514 A135515 A135516

Keyword

easy,nonn

Author

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

Status

approved