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.

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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..87.

Example

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].

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A136564 A136042 A268911 * A166044 A190287 A087707

Adjacent sequences: A351120 A351121 A351122 * A351124 A351125 A351126

Keyword

nonn,tabf

Author

Flávio V. Fernandes, Feb 01 2022

Extensions

Data corrected by Mohsen Maesumi, Jul 18 2022

Last row completed by Michel Marcus, Jul 18 2022

Status

approved