A166549 -id:A166549 - cp's OEIS frontend

A159966 Lodumo_4 of A102370 (sloping binary numbers).

0, 3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 19, 18, 17, 20, 23, 22, 21, 24, 27, 26, 25, 28, 31, 30, 29, 32, 35, 34, 33, 36, 39, 38, 37, 40, 43, 42, 41, 44, 47, 46, 45, 48, 51, 50, 49, 52, 55, 54, 53, 56, 59, 58, 57, 60, 63, 62, 61, 64, 67, 66, 65, 68, 71, 70, 69, 72

Offset: 0

Philippe Deléham, Apr 28 2009

A permutation of the nonnegative integers.
A092486 preceded by a zero. - Philippe Deléham, May 05 2009
Fixed points are the even numbers. - Wesley Ivan Hurt, Oct 16 2015

G. C. Greubel, Table of n, a(n) for n = 0..10000
OEIS wiki, Lodumo transform.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A005843, A102370, A132429, A158459, A166549.

[n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4) : n in [0..100]]; // Wesley Ivan Hurt, Oct 16 2015

A159966:=n->n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4): seq(A159966(n), n=0..100); # Wesley Ivan Hurt, Oct 16 2015

Table[n - (1 - (-1)^n) (-1)^((2 n + 1 - (-1)^n)/4), {n, 0, 40}] (* or *) CoefficientList[Series[(3 x - 4 x^2 + 3 x^3)/((x - 1)^2 (1 + x^2)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Oct 16 2015 *)
LinearRecurrence[{2,-2,2,-1},{0,3,2,1},80] (* Harvey P. Dale, Jul 02 2022 *)

concat(0, Vec((3*x-4*x^2+3*x^3)/((x-1)^2*(1+x^2)) + O(x^100))) \\ Altug Alkan, Oct 17 2015

a(n) = lod_4 (A102370(n)).
From Wesley Ivan Hurt, Oct 16 2015: (Start)
G.f.: (3*x-4*x^2+3*x^3)/((x-1)^2*(1+x^2)).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4), n>3.
a(n) = n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4).
a(2n) = A005843(n); a(2n+1) = A166549(n).
a(n+1) - a(n) = A132429(n)*(-1)^n. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023

A351122 Irregular triangle read by rows in which row n lists the number of divisions by 2 after tripling steps in the Collatz 3x+1 trajectory of 2n+1 until it reaches 1.

Original entry on oeis.org

1, 4, 4, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 1, 1, 1, 5, 4, 2, 3, 4, 1, 3, 1, 2, 3, 4, 6, 1, 1, 5, 4, 2, 1, 3, 1, 2, 3, 4, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4, 3, 1, 2, 3, 4

Offset: 1

Flávio V. Fernandes, Feb 01 2022

			Triangle starts at T(1,0):
   n\k   0   1   2   3   4   5   6   7   8 ...
   1:    1   4
   2:    4
   3:    1   1   2   3   4
   4:    2   1   1   2   3   4
   5:    1   2   3   4
   6:    3   4
   7:    1   1   1   5   4
   8:    2   3   4
   9:    1   3   1   2   3   4
  10:    6
  11:    1   1   5   4
  12:    2   1   3   1   2   3   4
  13:    1   2   1   1   1   1   2   2   1   2   1   1   2  ... (see A372362)
  ...
For n=6, the trajectory of 2*n+1 = 13 is as follows. The tripling steps ("=>") are followed by runs of 3 and then 4 halvings ("->"), so row n=6 is 3, 4.
  13  =>  40 -> 20 -> 10 -> 5  =>  16 -> 8 -> 4 -> 2 -> 1
    triple   \------------/   triple  \---------------/
               3 halvings                4 halvings
Runs of halvings are divisions by 2^T(n,k). Row n=11 is 1, 1, 5, 4 and its steps starting from 2*n+1 = 23 reach 1 by a nested expression
  (((((((23*3+1)/2^1)*3+1)/2^1)*3+1)/2^5)*3+1)/2^4 = 1.

Cf. A075680 (row lengths), A166549 (row sums), A351123 (row partial sums).
Cf. A256598.
Cf. A020988 (where row is [2*n]).
Cf. A198584 (where row length is 2), A228871 (where row is [1, x]).
Cf. A372362 (row 13, the first 41 terms).

row(n) = my(m=2*n+1, list=List()); while (m != 1, if (m%2, m = 3*m+1, my(nb = valuation(m,2)); m/=2^nb; listput(list, nb));); Vec(list); \\ Michel Marcus, Jul 18 2022

T(n,k) = log_2( (3*A256598(n,k)+1) / A256598(n,k+1) ).

Corrected by Michel Marcus, Jul 18 2022

A351123 Irregular triangle read by rows: row n lists the partial sums of the number of divisions by 2 after each tripling step in the Collatz trajectory of 2n+1.

Original entry on oeis.org

1, 5, 4, 1, 2, 4, 7, 11, 2, 3, 4, 6, 9, 13, 1, 3, 6, 10, 3, 7, 1, 2, 3, 8, 12, 2, 5, 9, 1, 4, 5, 7, 10, 14, 6, 1, 2, 7, 11, 2, 3, 6, 7, 9, 12, 16, 1, 3, 4, 5, 6, 7, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 23, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 48, 50, 52, 56, 59, 60, 61, 66, 70

Offset: 1

Flávio V. Fernandes, Feb 01 2022

The terms in row n are T(n,0), T(n,1), ..., T(n, A258145(n)-2), and are the partial sums of the terms in row n of A351122.
In each row n, the terms also satisfy the equation 3* (3* (3* (3* ... (3* (2n+1) +1) + 2^T(n,0)) + 2^T(n,1)) + 2^T(n,2)) + ... = 2^T(n, A258145(n)-2); e.g., for n=4, and A258145(4)-2=5: 3* (3* (3* (3* (3* (3*9+1) +2^2) +2^3) +2^4) +2^6) +2^9 = 2^13.
For row n, the right-hand side of the equation above is 2^A166549(n+1). E.g., for the above example (n=4), the right-hand side is 2^A166549(4+1) = 2^13.

			Triangle starts at T(1,0):
n\k   0   1   2   3   4   5   6   7   8   9   10 ...
1:    1   5
2:    4
3:    1   2   4   7  11
4:    2   3   4   6   9  13
5:    1   3   6  10
6:    3   7
7:    1   2   3   8   12
8:    2   5   9
...
E.g., row 3 of A351122 is [1, 1, 2, 3, 4]; its partial sums are [1, 2, 4, 7, 11].

Cf. A351122, A256598, A258145, A087230, A166549, A075680.

orow(n) = my(m=2*n+1, list=List()); while (m != 1, if (m%2, m = 3*m+1, my(nb = valuation(m,2)); m/=2^nb; listput(list, nb));); Vec(list); \\ A351122
row(n) = my(v = orow(n)); vector(#v, k, sum(i=1, k, v[i])); \\ Michel Marcus, Jul 18 2022

Data corrected by Mohsen Maesumi, Jul 18 2022

Last row completed by Michel Marcus, Jul 18 2022

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A159966 Lodumo_4 of A102370 (sloping binary numbers).

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A351122 Irregular triangle read by rows in which row n lists the number of divisions by 2 after tripling steps in the Collatz 3x+1 trajectory of 2n+1 until it reaches 1.

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A351123 Irregular triangle read by rows: row n lists the partial sums of the number of divisions by 2 after each tripling step in the Collatz trajectory of 2n+1.

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