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A263716 Irregular triangle read by rows: numbers in the Collatz conjecture in the order of their first appearance.

%I A263716 #41 Dec 15 2022 05:23:45
%S A263716 1,2,3,10,5,16,8,4,6,7,22,11,34,17,52,26,13,40,20,9,28,14,12,15,46,23,
%T A263716 70,35,106,53,160,80,18,19,58,29,88,44,21,64,32,24,25,76,38,27,82,41,
%U A263716 124,62,31,94,47,142,71,214,107,322,161,484,242,121,364,182
%N A263716 Irregular triangle read by rows: numbers in the Collatz conjecture in the order of their first appearance.
%C A263716 This is the irregular triangle read by rows giving trajectory of n in the Collatz problem, flattened and with all the repeated terms deleted.
%C A263716 This sequence goes to infinity as n gets larger. On the Collatz conjecture this sequence is a permutation of the positive integers. [Corrected by _Charles R Greathouse IV_, Jul 29 2016]
%H A263716 Daniel Suteu, <a href="/A263716/b263716.txt">Table of n, a(n) for n = 0..19999</a>
%H A263716 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A263716 row(n) = {
%F A263716    if seen[n]: stop
%F A263716    else: write(n) and do:
%F A263716    | n is one: stop
%F A263716    | n is odd: n <- 3*n+1
%F A263716    | n is even: n <- n/2
%F A263716 }
%e A263716 Triangle begins:
%e A263716 1;
%e A263716 2;
%e A263716 3, 10, 5, 16, 8, 4;
%e A263716 ...
%e A263716 The Collatz trajectories for the first five positive integers are {1}, {2, 1}, {3, 10, 5, 16, 8, 4, 2, 1}, {4, 2, 1}, {5, 16, 8, 4, 2, 1}.
%e A263716 From {2, 1} we delete 1 because it has already occurred. From {3, 10, 5, ..., 4, 2, 1} we delete {2, 1} because both numbers have already occurred. We completely get rid of {4, 2, 1} because it has already occurred as the tail end of {3, 10, 5, ...}, and we also completely get rid of {5, 16, 8, ...} for the same reason.
%e A263716 This leaves us with {1}, {2}, {3, 10, 5, 16, 8, 4}, thus accounting for the first eight terms of this sequence.
%t A263716 collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; DeleteDuplicates[Flatten[Table[collatz[n], {n, 20}]]] (* _Alonso del Arte_, Oct 24 2015 *)
%o A263716 (Sidef)
%o A263716 func collatz(n) is cached {  # automatically memoized function
%o A263716     say n;                   # prints the first unseen numbers
%o A263716     n.is_one ? 0
%o A263716              : (n.is_even ? collatz(n/2)
%o A263716                           : collatz(3*n + 1));
%o A263716 }
%o A263716 range(1, Math.inf).each { |i| collatz(i) }
%Y A263716 Cf. A006577, A070165, A222118 (row lengths).
%Y A263716 Cf. A347265 (essentially the same).
%K A263716 nonn,tabf,easy,nice
%O A263716 0,2
%A A263716 _Daniel Suteu_, Oct 24 2015