Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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