A324246 -id:A324246 - cp's OEIS frontend

A324039 Row lengths of the irregular triangles A324038 and A324246.

1, 1, 2, 2, 4, 8, 11, 19, 31, 50, 88, 146, 241, 407, 675, 1118, 1871, 3102, 5175, 8633, 14394, 23999, 40002, 66666, 111044, 185192, 308442, 514138, 857283, 1428537, 2381021

Offset: 0

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 09 2019

			a(5) = 8 = 2*a(4) because a_1(4) =  A324040(4) = 0.
a(6) =  2*a(5) - 5 = 16 - 5 = 11  because a_1(5) = A324040(5) = 5.

Cf. A324038, A324040, A324246.

a(n) = 2*a(n-1) - a_1(n-1), for n >= 2, and a(0) = 1 = a(1), where a_1(k) is the number of entries 1 (mod 3) in row k of A324246 given in A324040(k).

A324040 Number of vertex labels congruent to 1 modulo 3 of level n of the irregular triangle A324246.

Original entry on oeis.org

0, 0, 2, 0, 0, 5, 3, 7, 12, 12, 30, 51, 75, 139, 232, 365, 640, 1029, 1717, 2872, 4789, 7996, 13338, 22288, 36896, 61942, 102746, 170993, 286029, 476053, 793800

Offset: 0

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 09 2019

a(n) is also the number of vertex labels congruent to 3 modulo 6 of row n of the irregular triangle A324038.

This entry is interesting because it determines the number of vertices with out-degree 1 of level n, for n >= 1, of the modified reduced Collatz trees A324038 and A324246. All other vertices have out-degree 2. Hence this sequence determines recursively the number A324039(n) of vertices of label n of these two trees.

Cf. A324038, A324039, A324246.

a(n) = 2*A324039(n) - A324039(n-1), for n >= 1, and a(0) = 0. Implied by the definition of a(n) given in the name.

A324038 Irregular triangle T read by rows: Row n gives the vertex labels of level n of the tree related to the modified reduced Collatz map A324036.

Original entry on oeis.org

1, 5, 3, 21, 13, 85, 17, 53, 113, 341, 11, 69, 35, 213, 75, 453, 227, 1365, 7, 45, 277, 23, 141, 853, 301, 1813, 151, 909, 5461, 9, 29, 181, 369, 1109, 15, 93, 565, 1137, 3413, 401, 1205, 2417, 7253, 201, 605, 3637, 7281, 21845, 37, 19, 117, 241, 725, 1477, 739, 4437, 61, 373, 753, 2261, 4549, 2275, 13653, 267, 1605, 803, 4821, 1611, 9669, 4835, 29013, 805, 403, 2421, 4849, 14549, 29125, 14563, 87381

Offset: 0

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 09 2019

The length of row n is A324039, for n >= 0.
The branches of this incomplete binary tree, called CfsTree, give the iterations of the vertex labels of level n of the modified reduced Collatz map Cfs defined by Cfs(2*k+1) = A324036(k), for k >= 0, until at level n = 0 the label 1 is reached for the first time.
The out-degree of a vertex label T(n, k), for n >= 1, is 1 if T(n, k) == 3 (mod 6) and 2 for all other vertices. For level n = 0 with vertex label 1 this rule does not hold, it has out-degree 1, not 2.
The number of vertex labels on level n which are 3 (mod 6) is given by A324040(n).
The corresponding tree with nonnegative vertex labels t(n, k) = (T(n,k) - 1)/2 is given in A324246.
The Collatz conjecture is that all positive odd integers appear in this CfsTree. Because the sets of labels on the levels are pairwise disjoint these odd numbers will then appear just once.
For this tree see Figure 1 in the Vaillant-Delarue link. It is also shown in the W. Lang link.

			The irregular triangle T begins (the brackets combine pairs coming from out-degree 2 vertices of the preceding level):
n/k   1  2     3   4     5    6     7    8    9  10    11 ...
-------------------------------------------------------------
0:   1
1:   5
2:  (3  21)
3:  13  85
4:  (17 53) (113 341)
5:  (11 69)  (35 213)  (75  453) (227 1365)
6:  ( 7 45)  277 (23   141) 853   301 1813 (151 909) 5461
...
Row n = 7: (9 29) 181 (369 1109) (15 93)  565 (1137  3413) (401 1205) (2417 7253) (201 605) 3637 (7281 21845);
Row n = 8: 37 (19 117) (241 725) 1477 (739 4437) 61 373 (753 2261) 4549 (2275 13653) (267 1605) (803 4821) (1611 9669) (4835 29013) 805 (403 2421) (4849 14549) 29125 (14563 87381).
...
The successors of T(1,1) = 5 == 5 (mod 6) are (2*5 - 1)/3 = 3 and 4*5 + 1 = 21. The successor of T(2, 1) = 3 == 3 (mod 6) is 4*3 + 1 = 13. The successors of T(3, 1) = 13 == 1 (mod 6) are (4*13 - 1)/3 = 17 and 4*13 + 1 = 53.

Wolfdieter Lang, Collatz Trees from Vaillant-Delarue Maps
Nicolas Vaillant and Philippe Delarue, The hidden face of the 3x+1 problem. Part I: Intrinsic algorithm, April 26 2019.

Cf. A324036, A324039, A324040, A324246.

Recurrence: CfsTree(n), the list of vertex labels {T(n, k), for k = 1..A324038(n)} of level n, is obtained from:  CfsTree(0) = {1}, CfsTree(1) = {5}, and for n >= 2, CfsTree(n) = {2*m + 1 >= 1: fs(2*m+1) = T(n-1, k)), for k = 1..A324038(n-1)}, with fs from A324036.
Explicit form for the successor(s) of T(n, k) on level n+1, for n >= 1:
  a vertex label with T(n, k) == 3 (mod 6) produces the label 4*T(n, k) + 1 on level n+1; label T(n, k) == 1 (mod 6) produces the two labels (4*T(n, k) - 1)/3 and 4*T(n, k) + 1; label T(n, k) == 5 (mod 6) produces the two labels (2*T(n, k) - 1)/3 and 4*T(n, k) + 1.

A324245 The modified Collatz map considered by Vaillant and Delarue.

Original entry on oeis.org

0, 2, 0, 5, 3, 8, 1, 11, 6, 14, 2, 17, 9, 20, 3, 23, 12, 26, 4, 29, 15, 32, 5, 35, 18, 38, 6, 41, 21, 44, 7, 47, 24, 50, 8, 53, 27, 56, 9, 59, 30, 62, 10, 65, 33, 68, 11, 71, 36, 74, 12, 77, 39, 80, 13, 83, 42, 86, 14, 89, 45, 92, 15, 95, 48, 98, 16, 101, 51, 104, 17, 107, 54, 110, 18, 113, 57, 116, 19, 119, 60

Offset: 0

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 09 2019

This is a modified Collatz-Terras map (A060322), called in the Vaillant and Delarue link f.
The Collatz conjecture: iterations of the map f = a: N_0 -> N_0 with n -> a(n) lead always to 0.
The minimal number k with a^{[k]}(n) = 0 is given by A324037(n).
The tree CfTree, related to this map, giving the branches which lead to 0 for each vertex label of level n >= 0 is given in A324246.

Antti Karttunen, Table of n, a(n) for n = 0..20000
Nicolas Vaillant and Philippe Delarue, The hidden face of the 3x+1 problem. Part I: Intrinsic algorithm, April 26 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A060322, A324037, A324246, A349414.

a[n_]:=If[OddQ@n,(3n+1)/2,If[Mod[n,4]==0,3n/4,(n-2)/4]];Array[a,51,0] (* Giorgos Kalogeropoulos, Dec 08 2021 *)

A324245(n) = if(n%2, (1+3*n)/2, if(!(n%4), 3*(n/4), (n-2)/4)); \\ (After Mathematica-code) - Antti Karttunen, Dec 09 2021

a(n) = (3*n+1)/2 if n is odd, 3*n/4 if n == 0 (mod 4), and (n-2)/4 if n == 2 (mod 4).
a(n) = A349414(n) + n. - Ruud H.G. van Tol, Dec 08 2021
G.f.: x*(2 + 5*x^2 + 3*x^3 + 4*x^4 + x^5 + x^6)/(1 - x^4)^2. - Stefano Spezia, Dec 08 2021

More terms from Antti Karttunen, Dec 09 2021

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A324039 Row lengths of the irregular triangles A324038 and A324246.

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A324040 Number of vertex labels congruent to 1 modulo 3 of level n of the irregular triangle A324246.

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A324038 Irregular triangle T read by rows: Row n gives the vertex labels of level n of the tree related to the modified reduced Collatz map A324036.

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A324245 The modified Collatz map considered by Vaillant and Delarue.

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