A353149 - OEIS

%I #54 May 23 2022 21:44:32

%S 1,3,5,7,9,12,13,15,20,19,21,28,25,27,37,31,33,44,37,42,52,43,45,60,

%T 54,51,68,56,57,83,61,63,84,67,81,92,73,75,100,90,81,113,85,87,130,91,

%U 93,124,104,114,132,103,105,143,126,120,148,115,117,175,121,123,180,127,150,173,133,135,180,175

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

%C 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).

%C 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.

%C Sigma(n) <= a(n).

%C 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

%H David A. Corneth, <a href="/A353149/b353149.txt">Table of n, a(n) for n = 1..10000</a>

%H OEIS Plot 2, <a href="https://oeis.org/plot2a?name1=A353149&amp;name2=A000203&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=matp&amp;drawlines=true">A353149 vs A000203</a>

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

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

%e 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.

%e Illustration of a(15) = 37:

%e Level                                   Diagram

%e .                                          _

%e 1                                        _|1|_

%e 2                                      _|1   1|_

%e 3                                    _|1       1|_

%e 4                                  _|1           1|_

%e 5                                _|1       _       1|_

%e 6                              _|1        |1|        1|_

%e 7                            _|1          | |          1|_

%e 8                          _|1           _| |_           1|_

%e 9                        _|1            |1   1|            1|_

%e 10                     _|1              |     |              1|_

%e 11                   _|1               _|     |_               1|_

%e 12                 _|1                |1       1|                1|_

%e 13               _|1                  |         |                  1|_

%e 14             _|1                   _|    _    |_                   1|_

%e 15            |1                    |1    |1|    1|                    1|

%e .

%e The diagram has 37 steps, so a(15) = 37.

%o (PARI) a(n) = { my(r = A196020row(n)); sum(i = 0, (#r-1)\2, r[2*i + 1]) }

%o 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

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

%K nonn

%O 1,2

%A _Omar E. Pol_, Apr 26 2022