A238475 -id:A238475 - cp's OEIS frontend

A238477 a(n) = 32*n - 27 for n >= 1. Second column of triangle A238475.

5, 37, 69, 101, 133, 165, 197, 229, 261, 293, 325, 357, 389, 421, 453, 485, 517, 549, 581, 613, 645, 677, 709, 741, 773, 805, 837, 869, 901, 933, 965, 997, 1029, 1061, 1093, 1125, 1157, 1189, 1221, 1253, 1285, 1317, 1349, 1381, 1413, 1445, 1477, 1509, 1541, 1573

Offset: 1

Wolfdieter Lang, Mar 10 2014

This sequence gives all start numbers a(n) (sorted increasingly) of Collatz sequences of length 6 following the pattern udddd = ud^4, with u (for 'up'), mapping an odd number m to 3*m+1, and d (for 'down'), mapping an even number m to m/2. The last entry of this sequence is required to be odd and it is given by 6*n - 5.

This appears in Example 2.1. for x = 4 in the M. Trümper paper given as a link below.

			a(1) = 5 because the Collatz sequence of length 6 is [5, 16, 8, 4, 2, 1], following the pattern udddd, ending in 1, and 5 is the smallest start number following this pattern ending in an odd number.
a(2) = 37 with the length 6 Collatz sequence [37, 112, 56, 28, 14, 7] ending in 12 - 5 = 7, and this is the second smallest start number with this sequence pattern ending in an odd number.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017077 (first column), A238475, A239123 (third column).

CoefficientList[Series[(5 + 27 x)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 12 2014 *)

O.g.f.: x*(5+27*x)/(1-x)^2.
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 27 + exp(x)*(32*x - 27).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A239123 a(n) = 128*n - 107 for n >= 1. Third column of triangle A238475.

Original entry on oeis.org

21, 149, 277, 405, 533, 661, 789, 917, 1045, 1173, 1301, 1429, 1557, 1685, 1813, 1941, 2069, 2197, 2325, 2453, 2581, 2709, 2837, 2965, 3093, 3221, 3349, 3477, 3605, 3733, 3861, 3989, 4117, 4245, 4373, 4501, 4629, 4757, 4885, 5013, 5141, 5269, 5397, 5525, 5653

Offset: 1

Wolfdieter Lang, Mar 10 2014

This sequence gives all start numbers a(n) (sorted increasingly) of Collatz sequences of length 8 following the pattern ud^6 with u (for `up'), mapping an odd number m to 3*m+1, and d (for `down'), mapping an even number m to m/2. The last entry of this Collatz sequence is required to be odd, and it is given by 6*n - 5.

This appears in Example 2.1. for x = 6 in the M. Trümper paper given as a link below.

			a(1) = 21 because the Collatz sequence of length 8 is [21, 64, 32, 16, 8, 4, 2, 1] ending in 6*1-5 = 1, and 21 is the smallest positive number following this pattern udddddd ending in an odd number.
a(2) = 149 with the length 8 Collatz sequence [149, 448, 224, 112, 56, 28, 14, 7] ending in 6*2 - 5 = 7, and 149 is the second smallest start number following this pattern ud^6, ending in an odd number.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wolfdieter Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A238475, A238477 (second column).

CoefficientList[Series[(21 + 107 x)/(1 - x)^2, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 12 2014 *)

O.g.f.: x*(21+107*x)/(1-x)^2.
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 107 + exp(x)*(128*x - 107).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A017077 a(n) = 8*n + 1.

Original entry on oeis.org

1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 97, 105, 113, 121, 129, 137, 145, 153, 161, 169, 177, 185, 193, 201, 209, 217, 225, 233, 241, 249, 257, 265, 273, 281, 289, 297, 305, 313, 321, 329, 337, 345, 353, 361, 369, 377, 385, 393, 401, 409, 417, 425, 433

Offset: 0

N. J. A. Sloane

Cf. A007519 (primes), subsequence of A047522.
a(n-1), n >= 1, gives the first column of the triangle A238475 related to the Collatz problem. - Wolfdieter Lang, Mar 12 2014
First differences of A054552. - Wesley Ivan Hurt, Jul 08 2014
An odd number is congruent to a perfect square modulo every power of 2 iff it is in this sequence. Sketch of proof: Suppose the modulus is 2^k with k at least three and note that the only odd quadratic residue (mod 8) is 1. By application of difference of squares and the fact that gcd(x-y,x+y)=2 we can show that for odd x,y, we have x^2 and y^2 congruent mod 2^k iff x is congruent to one of y, 2^(k-1)-y, 2^(k-1)+y, 2^k-y. Now when we "lift" to (mod 2^(k+1)) we see that the degeneracy between a^2 and (2^(k-1)-a)^2 "breaks" to give a^2 and a^2-2^ka+2^(2k-2). Since a is odd, the latter is congruent to a^2+2^k (mod 2^(k+1)). Hence we can form every binary number that ends with '001' by starting modulo 8 and "lifting" while adding digits as necessary. But this sequence is exactly the set of binary numbers ending in '001', so our claim is proved. - Rafay A. Ashary, Oct 23 2016
For n > 3, also the number of (not necessarily maximal) cliques in the n-antiprism graph. - Eric W. Weisstein, Nov 29 2017
Bisection of A016813. - L. Edson Jeffery, Apr 26 2022

			Illustration of initial terms:
.                                          o       o       o
.                          o     o     o     o     o     o
.              o   o   o     o   o   o         o   o   o
.      o o o     o o o         o o o             o o o
.  o   o o o   o o o o o   o o o o o o o   o o o o o o o o o
.      o o o     o o o         o o o             o o o
.              o   o   o     o   o   o         o   o   o
.                          o     o     o     o     o     o
.                                          o       o       o
--------------------------------------------------------------
.  1       9          17              25                  33
- _Bruno Berselli_, Feb 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Antiprism Graph.
Eric Weisstein's World of Mathematics, Clique.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002189 (subsequence), A004768, A007519, A010731 (first differences), A016813, A047522, A054552.
Column 1 of A093565. Column 5 of triangle A130154. Second leftmost column of triangle A281334.
Row 1 of the arrays A081582, A238475, A371095, and A371096.
Row 2 of A257852.
Apart from the initial term, row sums of triangle A278480.

a017077 = (+ 1) . (* 8)
a017077_list = [1, 9 ..]  -- Reinhard Zumkeller, Dec 28 2012

I:=[1,9]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..60]]; // Vincenzo Librandi, Mar 14 2014

[8*n+1 : n in [0..50]]; // Wesley Ivan Hurt, Jul 08 2014

A017077:=n->8*n+1: seq(A017077(n), n=0..50); # Wesley Ivan Hurt, Jul 08 2014

Table[8 n + 1, {n, 0, 6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)
CoefficientList[Series[(1 + 7 x)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Mar 14 2014 *)
8 Range[0, 50] + 1 (* Wesley Ivan Hurt, Jul 08 2014 *)
LinearRecurrence[{2, -1}, {9, 17}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=8*n+1 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: (1+7*x)/(1-x)^2.
a(n+1) = A004768(n). - R. J. Mathar, May 28 2008
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Mar 14 2014
E.g.f.: exp(x)*(1 + 8*x). - Stefano Spezia, May 13 2021
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = a(n-1) + 8 with a(0)=1.
a(n) = A016813(2*n). (End)

A347834 An array A of the positive odd numbers, read by antidiagonals upwards, giving the present triangle T.

Original entry on oeis.org

1, 3, 5, 7, 13, 21, 9, 29, 53, 85, 11, 37, 117, 213, 341, 15, 45, 149, 469, 853, 1365, 17, 61, 181, 597, 1877, 3413, 5461, 19, 69, 245, 725, 2389, 7509, 13653, 21845, 23, 77, 277, 981, 2901, 9557, 30037, 54613, 87381, 25, 93, 309, 1109, 3925, 11605, 38229, 120149, 218453, 349525

Offset: 1

Wolfdieter Lang, Sep 20 2021

For the definition of this array A see the formula section.
The rows of A appear in a draft by Immmo O. Kerner in eqs. (1) and (2) as so-called horizontal sequences (horizontale Folgen). Thanks to Dr. A. Eckert for sending me this paper.
This array with entry A(k, n) becomes equal to the array T with T(n, k) given in A178415 by using a permutation of the rows, and changing the offset: A(k, n) = T(pe(k), n+1), with pe(3*(L+1)) = 4*(L+1), pe(1+3*L) = 1 + 2*L, pe(2+3*L) = 2*(1 + 2*L), for L >= 0. This permutation appears in A265667.
A proper sub-array is A238475(n, k) = A(1 + 3*(k-1), n-1), for k >= 1 and n >= 1.
In the directed Collatz tree with nodes labeled with only positive odd numbers (see A256598 for the paths), here called CTodd, the level L = 0 (on the top) has the node with label 1 as root. Because 1 -> 1 there is an arrow (a 1-cycle or loop) at the root. The level L = 1 consists of the nodes with labels A(1, n), for n >= 1, and each node is connected to 1 by a downwards directed arrow. The next levels for L >= 2 are obtained using the successor rule (used also by Kerner): S(u) = (4*u - 1)/3 if u == 1 (mod 3), (2*u - 1)/3 if u == 5 (mod 3), and there is no successor S(u) = empty if u = 3 (mod 6), that is, this node is a leaf.
However, each node with label u on level L >= 1, except a leaf, has as successors at level L + 1 not only the node with S(u) but all the nodes with labels A(S(u), n), for n >= 0.
In this way each node (also the root) of this CTodd has in-degree 1 and infinite out-degree (for L >= 2 there are infinitely many infinite outgoing arrows). All nodes with label A(k, n) with n >= 1, have the same precursor as the node A(k,0) in this tree for each k >= 1.
Except for the loop (1-cycle) for the root 1 there are no cycles in this directed tree CTodd.
That each number N = 5 + 8*K, for K >= 0 appears in array A for some column n >= 1 uniquely can be proved, using the fact of strictly increasing rows and columns, by showing that the columns n = 1, 2, ..., c contain all positive integers congruent to 5 modulo 8 except those of the positive congruence class A(1, c+1) modulo 2^(2*c+3) by induction on c. [added Dec 05 2021]
Row index k for numbers congruent to 5 modulo 8: Each number N = 5 + 8*K, for K >= 0, from A004770 is a member of row k of the array A starting with element A(k, 0) = (2*A065883(2 + 3*N) - 1)/3. For this surjective map see A347840. [simplified Dec 05 2021]
The Collatz conjecture can be reduced to the conjecture that in this rooted and directed tree CTodd each positive odd number appears as a label once, that is, all entries of the array A appear.

			The array A(k, n) begins:
k\n  0   1   2    3    4     5      6      7       8       9       10 ...
-------------------------------------------------------------------------
1:   1   5  21   85  341  1365   5461  21845   87381  349525  1398101
2:   3  13  53  213  853  3413  13653  54613  218453  873813  3495253
3:   7  29 117  469 1877  7509  30037 120149  480597 1922389  7689557
4:   9  37 149  597 2389  9557  38229 152917  611669 2446677  9786709
5:  11  45 181  725 2901 11605  46421 185685  742741 2970965 11883861
6:  15  61 245  981 3925 15701  62805 251221 1004885 4019541 16078165
7:  17  69 277 1109 4437 17749  70997 283989 1135957 4543829 18175317
8:  19  77 309 1237 4949 19797  79189 316757 1267029 5068117 20272469
9:  23  93 373 1493 5973 23893  95573 382293 1529173 6116693 24466773
10: 25 101 405 1621 6485 25941 103765 415061 1660245 6640981 26563925
...
--------------------------------------------------------------------
The triangle T(k, n) begins:
k\n  0  1   2    3    4     5     6      7      8      9 ...
------------------------------------------------------------
1:   1
2:   3  5
3:   7 13  21
4:   9 29  53   85
5:  11 37 117  213  341
6:  15 45 149  469 853   1365
7:  17 61 181  597 1877  3413  5461
8:  19 69 245  725 2389  7509 13653  21845
9:  23 77 277  981 2901  9557 30037  54613  87381
10: 25 93 309 1109 3925 11605 38229 120149 218453 349525
...
-------------------------------------------------------------
Row index k of array A, for entries 5 (mod 8).
213 = 5 + 8*26. K = 28 is even, (3*231+1)/16 = 40, A065883(40) = 10, hence A(k, 0) = N' = (10-1)/3 = 3, and k = 2. Moreover, n = log_4((3*213 + 1)/(3*A(2,0) + 1)) = log_4(64) = 3. 213 = A(2, 3).
85 = 5 + 8*10. K = 10 is even, (3*85 + 1)/16 = 16, A065883(16) = 1, N' = (1-1)/3 = 0 is even, hence A(k, 0) = 4*0 + 1 = 1, k = 1. 85 = A(1, 3).
61 = 5 + 8*7, K = 7 is odd, k = (7+1)/2 + ceiling((7+1)/4) = 6, and n = log_4((3*61 + 1)/(3*A(6,0) + 1)) = 1. 61 = A(6, 1).
----------------------------------------------------------------------------

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).
Immo O. Kerner, Elementarer Lösungsansatz zum Collatz-Ulam-Problem CUP oder das "3a + 1" Problem, Version Nov 26 2000.
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Row sequences of the array A, also diagonal sequences of the triangle T: -A007583 (k=0), A002450(n+1), A072197, A072261(n+1), A206374(n+1), A072262(n+1), A072262(n+1), A072201(n+1), A330246(n+1), ...
Column sequences of the array A, also of the triangle T (shifted): A047529, A347836, A347837, ...
Cf. A004770, A007494, A016777, A065883, A047395, A047529, A178415, A238475, A256598, A265667, A319451, A347839, A347840.

# Seen as an array:
A := (n, k) -> ((3*(n + floor(n/3)) - 1)*4^(k+1) - 2)/6:
for n from 1 to 6 do seq(A(n, k), k = 0..9) od;
# Seen as a triangle:
T := (n, k) -> 2^(2*k + 1)*(floor((n - k)/3) - k + n - 1/3) - 1/3:
for n from 1 to 9 do seq(T(n, k), k = 0..n-1) od;
# Using row expansion:
gf_row := k -> (1 / (x - 1) - A047395(k)) / (4*x - 1):
for k from 1 to 10 do seq(coeff(series(gf_row(k), x, 11), x, n), n = 0..10) od;
# Peter Luschny, Oct 09 2021

A347834[k_, n_] := (4^n*(6*(Floor[k/3] + k) - 2) - 1)/3;
Table[A347834[k - n, n], {k, 10}, {n, 0, k - 1}] (* Paolo Xausa, Jun 26 2025 *)

Array A:
A(k, 0) = A047529(k) (the positive odd numbers {1, 3, 7} (mod 8));
A(k, n) = ((3* A(k, 0) + 1)*4^n - 1)/3, for k >= 1 and n >= 0.
Recurrence for rows k >= 1: A(k, n) = 4*A(k, n-1) + 1, for n >= 1, with A(k, 0) = 2*(k + floor(k/3)) - 1 = A047529(k).
Explicit form: A(k, n) = ((3*(k + floor(k/3)) - 1)*4^(n+1) - 2)/6, k >= 1, n >= 0. Here 3*(k + floor(k/3)) = A319451(k).
Hence A(k, n) = 5 + 8*(2*A(k, n-2)), for n >= 1, with A(k, 0) = 2*(k + floor(k/3)) - 1 = A047529(k), and 2*A(k, -1) = (A(k, 1) - 5)/8 = k - 1 + floor(k/3) (equals index n of A(k, 1) in the sequence (A004770(n+1))_{n >= 0}). A(k, -1) is half-integer if k = A007494(m) = m + ceiling(m/2), for m >= 1, and A(k, -1) = 2*K if k = 1 + 3*K = A016777(K), for K >= 0.
O.g.f.: expansion in z gives o.g.f.s for rows k, also for k = 0: -A007583; expansion in x gives o.g.f.s for columns n.
  G(z, x) = (2*(-1 + 3*z + 3*z^2 + 7*z^3)*(1-x) - (1-4*x)*(1-z^3)) / (3*(1-x)*(1-4*x)*(1-z)*(1-z^3)).
Triangle T:
T(k, n) = A(k - n, n), for k >= 1 and n = 0..k-1.
A(k, n) = [x^n] (1/(x - 1) - A047395(k)) / (4*x - 1). - Peter Luschny, Oct 09 2021

A239126 Rectangular array showing the starting values M(n, k), k >= 1, for the Collatz operation (ud)^n, n >= 1, ending in an odd number, read by antidiagonals.

Original entry on oeis.org

3, 7, 7, 11, 15, 15, 15, 23, 31, 31, 19, 31, 47, 63, 63, 23, 39, 63, 95, 127, 127, 27, 47, 79, 127, 191, 255, 255, 31, 55, 95, 159, 255, 383, 511, 511, 35, 63, 111, 191, 319, 511, 767, 1023, 1023, 39, 71, 127, 223, 383, 639, 1023, 1535, 2047, 2047

Offset: 1

Wolfdieter Lang, Mar 13 2014

The companion array and triangle for the odd end numbers N(n, k) is given in A239127.
The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here (following the M. Trümper paper given in the link) denoted by u for 'up' and d for 'down': u m = 3*m+1, if m is odd, and d m = m/2 if m is even. The present array gives all start numbers M(n, k) for the Collatz word (ud)^n = s^n (s = ud is useful because, except for the one letter word u, at least one d follows a letter u), with n >= 1, and k >= 1. Such Collatz sequences have the maximal number of u's (grow fastest).
This rectangular array is M of Example 2.2. with x=y = n, n >= 1, of the M. Trümper reference, pp. 7-8, written as a triangle by taking NE-SW diagonals. The Collatz sequence starting with M(n, k) has length 2*n+1 for each k and it ends in the odd number N(n, k) given in A239127.
The first row sequences of the array M (columns of triangle TM) are A004767, A004771, A125169, A239128, ...

			The rectangular array M(n, k) begins:
n\k     1    2    3    4     5     6     7     8     9    10 ...
1:      3    7   11   15    19    23    27    31    35    39
2:      7   15   23   31    39    47    55    63    71    79
3:     15   31   47   63    79    95   111   127   143   159
4:     31   63   95  127   159   191   223   255   287   319
5:     63  127  191  255   319   383   447   511   575   639
6:    127  255  383  511   639   767   895  1023  1151  1279
7:    255  511  767 1023  1279  1535  1791  2047  2303  2559
8:    511 1023 1535 2047  2559  3071  3583  4095  4607  5119
9:   1023 2047 3071 4095  5119  6143  7167  8191  9215 10239
10:  2047 4095 6143 8191 10239 12287 14335 16383 18431 20479
...
The triangle TM(m, n) begins (zeros are not shown):
m\n   1    2     3     4     5     6      7      8      9    10 ...
1:    3
2:    7    7
3:   11   15    15
4:   15   23    31    31
5:   19   31    47    63    63
6:   23   39    63    95   127   127
7:   27   47    79   127   191   255    255
8:   31   55    95   159   255   383    511    511
9:   35   63   111   191   319   511    767   1023   1023
10:  39   71   127   223   383   639   1023   1535   2047  2047
...
---------------------------------------------------------------------
n=1, ud, k=1: M(1, 1) = 3 = TM(1, 1), N(1,1) = 5 with the Collatz sequence  [3, 10, 5] of length 3.
n=1, ud, k=2: M(1, 2) = 7 = TM(2, 1), N(1,2) = 11 with the Collatz sequence  [7, 22, 11] of length 3.
n=4, (ud)^4, k=2: M(4, 2) = 63 = TM(5, 4), N(4,2) = 323 with the Collatz sequence  [63, 190, 95, 286, 143, 430, 215, 646, 323] of length 9.
n=5, (ud)^5, k=1: M(5, 1) = 63 =  TM(5, 5), N(5,1) = 485 with the Collatz sequence  [63, 190, 95, 286, 143, 430, 215, 646, 323, 970, 485] of length 11.

Wolfdieter Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Eric Weisstein's World of Mathematics, Collatz Problem.
Wikipedia, Collatz Conjecture.

Cf. A006577, A139399, A112695, A238475, A238476, A004767, A004771, A125169, A239127, A239128.

The array: M(n, k) = 2^(n+1)*k - 1 for n >= 1 and k >= 1.
The triangle: TM(m, n) = M(n, m-n+1) = 2^(n+1)*(m-n+1) - 1 for m >= n >= 1 and 0 for m < n.
a(n) = 4*A087808(A130328(n-1)) - 1 (conjectured). - Christian Krause, Jun 15 2021

A238476 Rectangular array with all start numbers Mo(n, k), k >= 1, for the Collatz operation ud^(2*n-1), n >= 1, ending in an odd number, read by antidiagonals.

Original entry on oeis.org

3, 7, 13, 11, 29, 53, 15, 45, 117, 213, 19, 61, 181, 469, 853, 23, 77, 245, 725, 1877, 3413, 27, 93, 309, 981, 2901, 7509, 13653, 31, 109, 373, 1237, 3925, 11605, 30037, 54613, 35, 125, 437, 1493, 4949, 15701, 46421, 120149, 218453

Offset: 1

Wolfdieter Lang, Mar 10 2014

The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here denoted (with M. Trümper, see the link) by u for 'up' and d for 'down': u m = 3*m+1, if m is odd, and d m = m/2 if m is even. The present array gives all start numbers Mo(n, k), k >= 1, for Collatz sequences following the pattern (word) ud^(2*n-1), with n >= 1, ending in an odd number. This end number does not depend on n and it is given by No(k) = 6*k - 1. This Collatz sequence has length 1 + (1 + 2*n - 1) = 2*n + 1.
This rectangular array is Example 2.1. with x = 2*n-1, n >= 1, of the M. Trümper reference, pp. 4-5, written as a triangle by taking NE-SW diagonals. The case x = 2*n, n >= 1, for the word ud^(2*n) appears as array and triangle A238475.
The first rows of array Mo (columns of triangle To) are A004767, A082285, A239124, ...

			The rectangular array Mo(n, k) begins:
n\k      1        2        3        4        5        6        7        8        9        10 ...
1:       3        7       11       15       19       23       27       31       35        39
2:      13       29       45       61       77       93      109      125      141       157
3:      53      117      181      245      309      373      437      501      565       629
4:     213      469      725      981     1237     1493     1749     2005     2261      2517
5:     853     1877     2901     3925     4949     5973     6997     8021     9045     10069
6:    3413     7509    11605    15701    19797    23893    27989    32085    36181     40277
7:   13653    30037    46421    62805    79189    95573   111957   128341   144725    161109
8:   54613   120149   185685   251221   316757   382293   447829   513365   578901    644437
9:  218453   480597   742741  1004885  1267029  1529173  1791317  2053461  2315605   2577749
10: 873813  1922389  2970965  4019541  5068117  6116693  7165269  8213845  9262421  10310997
...
---------------------------------------------------------------------------------------------
The triangle To(m, n) begins (zeros are not shown):
m\n    1    2    3     4     5      6      7       8       9      10 ...
1:     3
2:     7   13
3:    11   29   53
4:    15   45  117   213
5:    19   61  181   469   853
6:    23   77  245   725  1877   3413
7:    27   93  309   981  2901   7509  13653
8:    31  109  373  1237  3925  11605  30037   54613
9:    35  125  437  1493  4949  15701  46421  120149  218453
10:   39  141  501  1749  5973  19797  62805  185685  480597  873813
...
n=1, ud, k=1: Mo(1, 1) = 3 = To(1, 1), No(1) = 5 with the Collatz sequence [3, 10, 5] of length 3.
n=1, ud, k=2: Mo(1, 2) = 7 = Te(2, 1), No(2) = 11 with the Collatz sequence [7, 22, 11] of length 3.
n=5, ud^9, k=2: Mo(5, 2) = 1877 = Te(6,5), No(2) = 11 with the Collatz sequence [1877, 5632, 2816, 1408, 704, 352, 176, 88, 44, 22, 11] of length 11.

W. Lang, On Collatz' Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Eric Weisstein's World of Mathematics, Collatz Problem.
Wikipedia, Collatz Conjecture.

Cf. A006577, A139399, A112695, A238475, A004767, A082285, A239124.

Mo(n, k) = 2^(2*n)*k - (2^(2*n-1)+1)/3 for n >= 1 and k >= 1.

To(m, n) = Mo(n, m-n+1) = 2^(2*n)*(m-n+1) - (2^(2*n-1)+1)/3 for m >= n >= 1 and 0 for m < n.

A240222 Rectangular array giving all start values M(n, k), k >= 1, for Collatz sequences following the pattern (udd)^(n-1) ud, n >= 1, read by antidiagonals.

Original entry on oeis.org

1, 3, 1, 5, 9, 1, 7, 17, 33, 1, 9, 25, 65, 129, 1, 11, 33, 97, 257, 513, 1, 13, 41, 129, 385, 1025, 2049, 1, 15, 49, 161, 513, 1537, 4097, 8193, 1, 17, 57, 193, 641, 2049, 6145, 16385, 32769, 1, 19, 65, 225, 769, 2561, 8193, 24577, 65537, 131073, 1, 21, 73, 257, 897, 3073, 10241, 32769, 98305

Offset: 1

Wolfdieter Lang, Apr 02 2014

The companion array and triangle for the end numbers N(n, k) is given in A240223.

The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here (following the M. Trümper paper given in the link) denoted by u for 'up' and d for 'down': u m = 3*m+1, if m is odd, and d m = m/2 if m is even. The present array gives all start numbers M(n, k) for Collatz sequences realizing the Collatz word (udd)^n ud = (sd)^n s (s = ud is useful because, except for the one letter word u, at least one d follows a letter u), with n >= 1, and k >= 1. The length of these Collatz sequences 3*n. For these Collatz sequences M(n, 0) = M(1, 0) = 1 and N(n, 0) = N(1, 0) = 2.

			The rectangular array M(n, k) begins:
n\k 0       1       2       3       4       5 ...
1:  1       3       5       7       9      11
2:  1       9      17      25      33      41
3:  1      33      65      97     129     161
4:  1     129     257     385     513     641
5:  1     513    1025    1537    2049    2561
6:  1    2049    4097    6145    8193   10241
7:  1    8193   16385   24577   32769   40961
8:  1   32769   65537   98305  131073  163841
9:  1  131073  262145  393217  524289  655361
10: 1  524289 1048577 1572865 2097153 2621441
...
For more columns see the link.
The triangle TM(m, n) begins (zeros are not shown):
k\n  1  2   3   4    5     6      7 ...
0:   1
1:   3  1
2:   5  9   1
3:   7 17  33   1
4:   9 25  65 129    1
5:  11 33  97 257  513     1
6:  13 41 129 385 1025  2049      1
...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1 = TM(0, 1), N(1, 0) = 2 with the Collatz sequence [1, 4, 2] of
length 3.
n=1, ud, k=2: M(1, 2) = 5 = TM(2, 1), N(1,  2) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1 = TM(1, 2), Ne(2, 0) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.

Wolfdieter Lang, Rectangular array and triangle.
Wolfdieter Lang, On Collatz' Words, Sequences and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
Eric Weisstein's World of Mathematics, Collatz Problem.
Wikipedia, Collatz Conjecture

Cf. A238475, A238476, A239126, A239127.

The array: M(n, k) = 1 + 2^(2*n-1)*k for n >= 1 and k >= 0.

The triangle: TM(m, n) = M(n,m-n+1) = 1 + 2^(2*n-1)*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n.

A240223 Rectangular companion array to M(n,k), given in A240222, showing the end numbers N(n, k), k >= 1, for the Collatz operation (udd)^(n-1) ud, n >= 1, read by antidiagonals.

Original entry on oeis.org

2, 5, 2, 8, 11, 2, 11, 20, 29, 2, 14, 29, 56, 83, 2, 17, 38, 83, 164, 245, 2, 20, 47, 110, 245, 488, 731, 2, 23, 56, 137, 326, 731, 1460, 2189, 2, 26, 65, 164, 407, 974, 2189, 4376, 6563, 2, 29, 74, 191, 488, 1217, 2918, 6563, 13124, 19685, 2, 32, 83, 218, 569, 1460, 3647, 8750, 19685, 39368, 59051, 2

Offset: 0

Wolfdieter Lang, Apr 04 2014

The companion array and triangle for the start numbers M(n, k) is given in A240222.

For the Collatz operations u (for 'up') and d (for 'down') see the comment on A240222, also for links, especially for the M. Trümper paper.

			The rectangular array N(n, k) begins
  n\k 0      1       2       3       4       5 ...
  1:  2      5       8      11      14      17
  2:  2     11      20      29      38      47
  3:  2     29      56      83     110     137
  4:  2     83     164     245     326     407
  5:  2    245     488     731     974    1217
  6:  2    731    1460    2189    2918    3647
  7:  2   2189    4376    6563    8750   10937
  8:  2   6563   13124   19685   26246   32807
  9:  2  19685   39368   59051   78734   98417
  10: 2  59051  118100  177149  236198  295247
  ...
For more columns see the link.
The triangle TN(m, n) begins (zeros are not shown):
  m\n  1  2   3   4    5    6    7 ...
  0:   2
  1:   5  2
  2:   8 11   2
  3:  11 20  29   2
  4:  14 29  56  83    2
  5:  17 38  83 164  245    2
  6:  20 47 110 245  488  731    2
  ...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1, N(1, 0) = TN(0, 1) = 2 with the Collatz sequence [1, 4, 2] of length 3.
n=1, ud, k=2: M(1, 2) = 5, N(1, 2) = TN(2, 1) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1, Ne(2, 0) = TN(1, 2) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.

Wolfdieter Lang, Rectangular array and triangle.
Wolfdieter Lang, On Collatz Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.

Cf. A238475, A238476, A239126, A239127, A240222, A016789 (first row of N), A017185 (second row of N).

The array: N(n, k) = 2 + 3^n*k for n >= 1 and k >= 0.

The triangle: TN(m, n) = N(n,m-n+1) = 2 + 3^n*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n.

cp's OEIS Frontend

A238477 a(n) = 32*n - 27 for n >= 1. Second column of triangle A238475.

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A239123 a(n) = 128*n - 107 for n >= 1. Third column of triangle A238475.

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A017077 a(n) = 8*n + 1.

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A347834 An array A of the positive odd numbers, read by antidiagonals upwards, giving the present triangle T.

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A239126 Rectangular array showing the starting values M(n, k), k >= 1, for the Collatz operation (ud)^n, n >= 1, ending in an odd number, read by antidiagonals.

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A238476 Rectangular array with all start numbers Mo(n, k), k >= 1, for the Collatz operation ud^(2*n-1), n >= 1, ending in an odd number, read by antidiagonals.

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A240222 Rectangular array giving all start values M(n, k), k >= 1, for Collatz sequences following the pattern (udd)^(n-1) ud, n >= 1, read by antidiagonals.

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A240223 Rectangular companion array to M(n,k), given in A240222, showing the end numbers N(n, k), k >= 1, for the Collatz operation (udd)^(n-1) ud, n >= 1, read by antidiagonals.

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