A353149 Sum of the odd-indexed terms in the n-th row of the triangle A196020.

1, 3, 5, 7, 9, 12, 13, 15, 20, 19, 21, 28, 25, 27, 37, 31, 33, 44, 37, 42, 52, 43, 45, 60, 54, 51, 68, 56, 57, 83, 61, 63, 84, 67, 81, 92, 73, 75, 100, 90, 81, 113, 85, 87, 130, 91, 93, 124, 104, 114, 132, 103, 105, 143, 126, 120, 148, 115, 117, 175, 121, 123, 180, 127, 150, 173, 133, 135, 180, 175

Offset

1,2

Comments

a(n) is the total number of steps in all odd-indexed double-staircases of the diagram of A196020 with n levels (see the example).

a(n) is also the total number of steps in all odd-indexed double-staircases of the diagram described in A335616 with n levels that have at least one step in the bottom level of the diagram.

Sigma(n) <= a(n).

The graph of the sum-of-divisors function A000203 is intermediate between the graph of this sequence and the graph of A353154 (see link). - Omar E. Pol, May 13 2022

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

OEIS Plot 2, A353149 vs A000203

Formula

a(n) = A000203(n) + A353154(n).

a(n) = A209246(n) - A353154(n).

Example

For n = 15 the 15th row of the triangle A196020 is [29, 13, 7, 0, 1]. The sum of the odd-indexed terms is 29 + 7 + 1 = 37, so a(15) = 37.
Illustration of a(15) = 37:
Level                                   Diagram
.                                          _
1                                        _|1|_
2                                      _|1   1|_
3                                    _|1       1|_
4                                  _|1           1|_
5                                _|1       _       1|_
6                              _|1        |1|        1|_
7                            _|1          | |          1|_
8                          _|1           _| |_           1|_
9                        _|1            |1   1|            1|_
10                     _|1              |     |              1|_
11                   _|1               _|     |_               1|_
12                 _|1                |1       1|                1|_
13               _|1                  |         |                  1|_
14             _|1                   _|    _    |_                   1|_
15            |1                    |1    |1|    1|                    1|
.
The diagram has 37 steps, so a(15) = 37.

Prog

(PARI) a(n) = { my(r = A196020row(n)); sum(i = 0, (#r-1)\2, r[2*i + 1]) }
A196020row(n) = { my(res, qc); qc = (sqrtint(8*n + 1) - 1)\2; res = vector(qc); for(i = 1, qc, cn = n - binomial(i + 1, 2); if(cn % i == 0, res[i] = 2*(cn/i) + 1 ) ); res } \\ David A. Corneth, Apr 28 2022

Crossrefs

Cf. A000203, A057640, A196020, A209246, A211343, A235791, A236104, A237591, A237593, A285901, A335616, A347186, A353154.

Sequence in context: A340439 A131628 A079091 * A191749 A038663 A291154

Adjacent sequences: A353146 A353147 A353148 * A353150 A353151 A353152

Keyword

nonn

Author

Omar E. Pol, Apr 26 2022

Status

approved