# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a324246 Showing 1-1 of 1 %I A324246 #15 Aug 10 2023 13:13:01 %S A324246 0,2,1,10,6,42,8,26,56,170,5,34,17,106,37,226,113,682,3,22,138,11,70, %T A324246 426,150,906,75,454,2730,4,14,90,184,554,7,46,282,568,1706,200,602, %U A324246 1208,3626,100,302,1818,3640,10922,18,9,58,120,362,738,369,2218,30,186,376,1130,2274,1137,6826,133,802,401,2410,805,4834,2417,14506,402,201,1210,2424,7274,14562,7281,43690 %N A324246 Irregular triangle T read by rows: T(n, k) = (A324038(n, k) - 1)/2. %C A324246 The length of row n is A324039, for n >= 0. %C A324246 This is the incomplete binary tree corresponding to the modified Collatz map f (from the Vaillant and Delarue link) given in A324245. %C A324246 The branches of this tree, called CfTree, give the iterations under the Vaillant and Delarue map f of the vertex labels of level n until label 0 on level n = 0 is reached. %C A324246 The out-degree of a vertex label T(n, k), for n >= 1, is 1 if T(n, k) == 1 (mod 3) and 2 for all other labels. For level n = 0 with vertex label 0 this rule does not hold, it has out-degree 1, not 2. %C A324246 The number of vertex labels on level n which are 1 (mod 3) is given in A324040. %C A324246 The corresponding tree CfsTree with only odd vertex labels t(n, k) = 2*T(n,k) + 1 is given in A324038. %C A324246 The Collatz conjecture is that all nonnegative integers appear in this CfTree. Because the sets of labels on the levels are pairwise disjoint, these numbers will then appear just once. %C A324246 For this tree see Figure 2 in the Vaillant-Delarue link. It is also shown in the W. Lang link given in A324038. %H A324246 Nicolas Vaillant and Philippe Delarue, The hidden face of the 3x+1 problem. Part I: Intrinsic algorithm, April 26 2019. %F A324246 Recurrence for the set of vertex labels CfTree(n) = {T(n, k), k = 1..A324039(n)} on level (row) n: %F A324246 This set is obtained, with the map f from A324245, from CfTree(0) = {0}, CfTree(1) = {2}, and for n >= 2 CfTree(n) = {m >= 0: f(m) = T(n-1, k), for k = 1.. A324039(n-1)}. %F A324246 Explicit form for the successor of T(n, k) on row (level) n+1, for n >= 1: %F A324246 a label with T(n, k) == 1 (mod 3) produces the label 2*(1 + 2*T(n, k)) on row n+1; label T(n, k) == 0 (mod 3) produces the two labels 4*T(n, k)/3 and 2*(1 + 2*T(n, k)); label T(n, k) == 2 (mod 3) produces the two labels (-1 + 2*T(n, k))/3 and 2*(1 + 2*T(n, k)). %e A324246 The irregular triangle T begins (the brackets combine pairs coming from out-degree 2 vertices of the preceding level): %e A324246 ---------------------------------------------------------- %e A324246 n\k 1 2 3 4 5 6 7 8 9 10 11 ... %e A324246 0: 0 %e A324246 1: 2 %e A324246 2: (1 10) %e A324246 3: 6 42 %e A324246 4: (8 26) (56 170) %e A324246 5: (5 34) (17 106) (37 226) (113 682) %e A324246 6: (3 22) 138 (11 70) 426 150 906 (75 454) 2730 %e A324246 ... %e A324246 Row n = 7: (4 14) 90 (184 554) (7 46) 282 (568 1706) (200 602) (1208 3626) (100 302) 1818 (3640 10922); %e A324246 Row n = 8: 18 (9 58) (120 362) 738 (369 2218) 30 186 (376 1130) 2274 (1137 6826) (133 802) (401 2410) (805 4834) (2417 14506) 402 (201 1210) (2424 7274) 14562 (7281 43690). %e A324246 ... %e A324246 The successors of T(1,1) = 2 == 2 (mod 3) are (-1 + 2*2 )/3 = 1 and 2*(1 + 2*2) = 10. The successor of T(2, 1) = 1 == 1 (mod 3) is 2*(1 + 2*1) = 6. The successors of T(3, 1) = 6 == 0 (mod 3) are 4*6/3 = 8 and 2*(1 + 2*6) = 26. %Y A324246 Cf. A248573 (Collatz-Terras tree), A324038 (CfsTree), A324039, A324040, A324245. %K A324246 nonn,tabf,easy %O A324246 0,2 %A A324246 _Nicolas Vaillant_, Philippe Delarue, _Wolfdieter Lang_, May 09 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE