A209246 -id:A209246 - cp's OEIS frontend

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

Omar E. Pol, Apr 26 2022

nonn

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

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000
OEIS Plot 2, A353149 vs A000203

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

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

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

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

A353154 Sum of the even-indexed terms in the n-th row of the triangle A196020.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 5, 0, 7, 1, 9, 0, 11, 3, 13, 0, 15, 5, 17, 0, 20, 7, 21, 0, 23, 9, 28, 0, 27, 11, 29, 0, 36, 13, 33, 1, 35, 15, 44, 0, 39, 17, 41, 3, 52, 19, 45, 0, 47, 21, 60, 5, 51, 23, 54, 0, 68, 25, 57, 7, 59, 27, 76, 0, 66, 29, 65, 9, 84, 31, 69, 0, 71, 33, 97, 11, 75, 36, 77, 0

Offset: 1

Omar E. Pol, Apr 27 2022

nonn

Conjecture: indices of zeros give A082662.
a(n) is the total number of steps in all even-indexed double-staircases of the diagram of A196020 with n levels.
a(n) is also the total number of steps in all even-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.
The graph of the sum-of-divisors function A000203 is intermediate between the graph of A353149 and the graph of this sequence (see the Links section). - Omar E. Pol, May 13 2022

			For n = 15 the 15th row of the triangle A196020 is [29, 13, 7, 0, 1]. The sum of the even-indexed terms is 13 + 0 = 13, so a(15) = 13.

OEIS Plot 2, A353154 vs A000203

Cf. A000203, A057641, A082662, A196020, A209246, A211343, A235791, A236104, A237591, A237593, A285901, A335616, A347186, A353149.

a(n) = A353149(n) - A000203(n).

a(n) = A209246(n) - A353149(n).

cp's OEIS Frontend

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

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