A324036 -id:A324036 - cp's OEIS frontend

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.

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.

A324037 The minimal number of iterations to reach 1 of the modified reduced Collatz function, defined for odd numbers 1 + 2*n in A324036 (assuming the Collatz conjecture).

Original entry on oeis.org

0, 2, 1, 6, 7, 5, 3, 7, 4, 8, 2, 6, 9, 48, 7, 46, 10, 5, 8, 14, 47, 11, 6, 45, 9, 10, 4, 49, 12, 13, 8, 47, 10, 11, 5, 44, 50, 5, 9, 15, 9, 48, 3, 12, 12, 40, 7, 46, 51, 10, 10, 38, 16, 43, 49, 30, 4, 13, 8, 14, 41, 19, 47, 20, 52, 11, 11, 16, 39, 17, 6

Offset: 0

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

nonn

The Collatz conjecture is that a(n) is finite. If 1 should never be reached then a(n) = -1.
Compare this sequence with the analogous one A075680(n+1) for the reduced Collatz map of A075677.
a(n) gives also the minimal number of iterations of the Vaillant-Delarue map f, defined in A324245, acting on n to reach 0 (assuming the Collatz conjecture).
For the link to the Vaillant-Delarue paper (where fs is called f_s) see A324036.

			a(4) = 7 because 1 + 2*4 = 9 and the 7 fs iterations acting on 9 are 7, 11, 17, 13, 3, 5, 1.
Compare this to the reduced Collatz map given in A075677 which needs only 6 = A075680(5) iterations 7, 11, 17, 13, 5, 1. The additional step in the fs case follows 13 == 5 mod(8).

Cf. A075677, A075680(n+1), A324036, A324245.

fs^[a(n)](1 + 2*n) = 1 but fs^[a(n)-1](1 + 2*n) is not 1 (for all n with finite a(n)), where fs is the modified reduced Collatz map defined for 1 + 2*n in A324036(n), for n >= 1, and a(0) = 0.

A072197 a(n) = 4*a(n-1) + 1 with a(0) = 3.

Original entry on oeis.org

3, 13, 53, 213, 853, 3413, 13653, 54613, 218453, 873813, 3495253, 13981013, 55924053, 223696213, 894784853, 3579139413, 14316557653, 57266230613, 229064922453, 916259689813, 3665038759253, 14660155037013, 58640620148053, 234562480592213, 938249922368853, 3752999689475413

Offset: 0

N. Rathankar (rathankar(AT)yahoo.com), Jul 03 2002

Let A be the Hessenberg matrix of order n, defined by: A[1, j] = 1, A[i, i] := 2, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n - 1) = (-1)^n*charpoly(A, -2). - Milan Janjic, Jan 26 2010
Numbers whose binary representation is 11 together with n times 01. For example, 213 = 11010101 (2). - Omar E. Pol, Nov 22 2012
The Collatz-function starting with a(n) will terminate at 1 after 2*n + 7 steps. This is because 3*a(n) + 1 = 5*2^(2n + 1), and the Collatz-function starting with 5 terminates at 1 after 5 additional steps.  So for example, a(2) = 53; Collatz sequence starting with 53 follows: 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 (11 steps). - Bob Selcoe, Apr 03 2015
a(n) is also the sum of the numerator and denominator of the binary fractions 0.1, 0.101, 0.10101, 0.1010101... Thus 0.1 = 1/2 with 1 + 2 = 3, 0.101 = 1/2 + 1/8 = 5/8 with 5 + 8 = 13; 0.10101 = 1/2 + 1/8 + 1/32 = 21/32 with 21 + 32 = 53. - J. M. Bergot, Sep 28 2016
a(n), for n >= 2, is also the smallest odd number congruent to 5 modulo 8 for which the modified reduced Collatz map given in A324036 has n consecutive extra steps compared to the reduced Collatz map given in A075677. - Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 09 2019

			a(1) = 13 because a(0) = 3 and 4 * 3 + 1 = 13.
a(2) = 53 because a(1) = 13 and 4 * 13 + 1 = 53.
a(3) = 213 because a(2) = 53 and 4 * 53 + 1 = 213.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Petro Kosobutskyy and Volodymyr Karkulovskyy, Recurrence and structuring of sequences of transformations 3n+ 1 as arguments for confirmation of the Collatz hypothesis, Comp. Des. Sys., Theor. Practice (2023) Vol. 5, No. 1, pp. 28-33. See p. 29.
Petro Kosobutskyy, Anastasiia Yedyharova, and Taras Slobodzyan, From Newton's binomial and Pascal's triangle to Collatz's problem, Comp. Des. Sys., Theor. Practice (2023) Vol. 5, No. 1, pp. 121-127.
Index entries for linear recurrences with constant coefficients, signature (5,-4).
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A000302, A002066 (first differences), A002450, A007583, A020988, A075677, A178415, A324036, A347834.

[(10*4^n-1)/3: n in [0..30]]; // Vincenzo Librandi, Oct 31 2011

A072197:=n->(10*4^n - 1)/3: seq(A072197(n), n=0..30); # Wesley Ivan Hurt, Sep 29 2016

Table[(10(4^n) - 1)/3, {n, 0, 19}] (* Alonso del Arte, Nov 22 2012 *)
NestList[4#+1&,3,30] (* Harvey P. Dale, Mar 09 2019 *)

a(n)=10*4^n\3 \\ Charles R Greathouse IV, Apr 06 2016

Vec((3-2*x)/((1-x)*(1-4*x)) + O(x^30)) \\ Colin Barker, Sep 28 2016

a(n) = (10*4^n - 1)/3 = 10*A002450(n) + 3. - Henry Bottomley, Dec 02 2002
a(n) = 5*a(n-1) - 4*a(n-2), n > 1. - Vincenzo Librandi, Oct 31 2011
a(n) = 2^(2*(n + 1)) - (2^(2*n + 1) + 1)/3 = A000302(n + 1) - A007583(n). - Vladimir Pletser, Apr 12 2014
a(n) = (5*2^(2*n + 1) - 1)/3. - Bob Selcoe, Apr 03 2015
G.f.: (3-2*x)/((1-x)*(1-4*x)). - Colin Barker, Sep 28 2016
a(n) = A020988(n) + A020988(n+1) + 1 = 2*(A002450(n) + A002450(n+1)) + 1. - Yosu Yurramendi, Jan 24 2017
a(n) = A002450(n+1) + 2^(2*n+1). - Adam Michael Bere, May 13 2021
a(n) = a(n-1) + 5*2^(2*n-1), for n >= 1, with a(0) = 3. - Wolfdieter Lang, Aug 16 2021
a(n) = A178415(2,n+1) = A347834(2,n), arrays, for n >= 0. - Wolfdieter Lang, Nov 29 2021
E.g.f.: exp(x)*(10*exp(3*x) - 1)/3. - Elmo R. Oliveira, Apr 02 2025

More terms from Henry Bottomley, Dec 02 2002

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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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A324037 The minimal number of iterations to reach 1 of the modified reduced Collatz function, defined for odd numbers 1 + 2*n in A324036 (assuming the Collatz conjecture).

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A072197 a(n) = 4*a(n-1) + 1 with a(0) = 3.

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