%I
%S 1,3,36,6,20,72,1168,11,216,35,576,143,111,2422,1657,19,336,378,6253,
%T 66,51,1167,820,241,24096,180,18805,215,3833,3488,368905,31,3460,575,
%U 426,716,576,12387,57556,110,10513,83,8948,2303,1782,1656,175195,387,1647
%N Start from the singleton set S = {n}, and unless 1 is already a member of S, generate on each iteration a new set where each odd number k is replaced by 3k+1, and each even number k is replaced by 3k+1 and k/2. a(n) is the total size of the set from the singleton through after the first iteration which has produced 1 as a member, inclusive.
%C See comments at A290100.
%C A290100(n) is the size of the set at the last iteration, while this sequence is the sum of sizes of all generations including the last iteration.
%C A290100(n)/A291213(n) < 29/90 for n = {6, 67, 81, 92, 102, 153, 155, 165, 198, 201, 202, 204, 205, 217, 228, 235, 264, 265, 289, 299, 308, 309, 349, 353, 360, 396, 408, 434, ...}, with n = 6 the greatest observed difference.  _Michael De Vlieger_, Aug 30 2017
%H Michael De Vlieger, <a href="/A291213/b291213.txt">Table of n, a(n) for n = 1..450</a>
%e For n = 5:
%e Generation Set
%e 1 (1 term) 5
%e 2 (1 term) 16
%e 3 (2 terms) 8, 49
%e 4 (3 terms) 4, 25, 148
%e 5 (5 terms) 2, 13, 74, 76, 445
%e 6 (8 terms) 1, 7, 37, 38, 40, 223, 229, 1336
%e thus a(5) = 20 and A290100(5) = 8.
%t Table[Length@ Flatten@ NestWhileList[Union@ Flatten[# /. {k_ /; OddQ@ k :> 3 k + 1, k_ /; EvenQ@ k :> {k/2, 3 k + 1}}] &, {n}, FreeQ[#, 1] &], {n, 49}]
%Y Cf. A006577, A127885, A127886, A290100.
%K nonn
%O 1,2
%A _Michael De Vlieger_, Aug 26 2017
