A135511 - OEIS

%I #5 Feb 13 2024 23:16:11

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>.

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

%e 5 = 3(1) + 2 -> 2(2) + 1 -> 1(5) + 0 or

%e 5 = 3(1) + 2 -> 2(3) - 1 -> 1(5) + 0 or

%e 5 = 3(2) - 1 -> 1(5) + 0

%e So 3/5 = 1 - 1/2 + 1/10, 1 - 1/3 - 1/15, 1/2 + 1/10: thus h(5)=3.

%K easy,nonn

%O 3,2

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