A322141 -id:A322141 - cp's OEIS frontend

A240542 The midpoint of the (rotated) Dyck path from (0, n) to (n, 0) defined by A237593 has coordinates (a(n), a(n)). Also a(n) is the alternating sum of the n-th row of A235791.

1, 2, 2, 3, 3, 5, 5, 6, 7, 7, 7, 9, 9, 9, 11, 12, 12, 13, 13, 15, 15, 15, 15, 17, 18, 18, 18, 20, 20, 22, 22, 23, 23, 23, 25, 26, 26, 26, 26, 28, 28, 30, 30, 30, 32, 32, 32, 34, 35, 36, 36, 36, 36, 38, 38, 40, 40, 40, 40, 42, 42, 42, 44, 45, 45, 47, 47, 47, 47, 49, 49, 52, 52

Offset: 1

Hartmut F. W. Hoft, Apr 07 2014

nonn

The sequence is closely related to the alternating harmonic series.
Its asymptotic behavior is lim_{k -> infinity} a(k)/k = log 2. The relative error is abs(a(k) - k*log(2))/(k*log(2)) <= 2/sqrt(k).
Conjecture 1: the sequence of first positions of the alternating runs of odd and even numbers in a(k) equals A028982. Example: the positions in (1),(2),2,(3),3,5,5,(6),(7),7,7,9,9,9,11,(12),12,(13),13,15,... are 1,2,4,8,9,16,18,... Checked with a Mathematica function through a(1000000).
See A235791, A237591 and A237593 for additional formulas and properties.
Conjecture 2: The sequence of differences a(n) - a(n-1), for n>=1, appears to be equal to A067742(n), the sequence of middle divisors of n; as an empty sum, a(0) = 0, (which was conjectured by Michel Marcus in the entry A237593). Checked with the two respective Mathematica functions up to n=100000. - Hartmut F. W. Hoft, Jul 17 2014
The number of occurrences of n is A259179(n). - Omar E. Pol, Dec 11 2016
Conjecture 3: a(n) is also the difference between the total number of partitions of all positive integers <= n into an odd number of consecutive parts, and the total number of partitions of all positive integers <= n into an even number of consecutive parts. - Omar E. Pol, Apr 28 2017
Conjecture 4: a(n) is also the total number of central subparts of all symmetric representations of sigma of all positive integers <= n. - Omar E. Pol, Apr 29 2017
a(n) is also the sum of the odd-indexed terms of the n-th row of the triangle A237591. - Omar E. Pol, Jun 20 2018
a(n) is the total number of middle divisors of all positive integers <= n (after Michel Marcus's conjecture in A237593). - Omar E. Pol, Aug 18 2021

			From _Omar E. Pol_, Dec 22 2020: (Start)
Illustration of initial terms in two ways in accordance with the sum of the odd-indexed terms of the rows of A237591:
.
n   a(n)                               _              _
1    1                               _|_|            |_|_
2    2                             _|_ _|            |_ _|
3    2                           _|_ _|              |_ _|_
4    3                         _|_ _ _|              |_ _ _|
5    3                       _|_ _ _|  _             |_ _ _|_ _
6    5                     _|_ _ _ _| |_|            |_ _ _ _|_|
7    5                   _|_ _ _ _|   |_|            |_ _ _ _|_|_
8    6                 _|_ _ _ _ _|  _|_|            |_ _ _ _ _|_|_
9    7               _|_ _ _ _ _|   |_ _|            |_ _ _ _ _|_ _|
10   7             _|_ _ _ _ _ _|   |_|              |_ _ _ _ _ _|_|
11   7           _|_ _ _ _ _ _|    _|_|              |_ _ _ _ _ _|_|_ _
12   9         _|_ _ _ _ _ _ _|   |_ _|              |_ _ _ _ _ _ _|_ _|
13   9       _|_ _ _ _ _ _ _|     |_ _|              |_ _ _ _ _ _ _|_ _|
14   9     _|_ _ _ _ _ _ _ _|    _|_|  _             |_ _ _ _ _ _ _ _|_|_ _
15  11   _|_ _ _ _ _ _ _ _|     |_ _| |_|            |_ _ _ _ _ _ _ _|_ _|_|_
16  12  |_ _ _ _ _ _ _ _ _|     |_ _| |_|            |_ _ _ _ _ _ _ _ _|_ _|_|
...
                  Figure 1.                                  Figure 2.
.
Figure 1 shows the illustration of initial terms taken from the isosceles triangle of A237593. For n = 16 there are (9 + 2 + 1) = 12 cells in the 16th row of the diagram, so a(16) = 12.
Figure 2 shows the illustration of initial terms taken from an octant of the pyramid described in A244050 and A245092. For n = 16 there are (9 + 2 + 1) = 12 cells in the 16th row of the diagram, so a(16) = 12.
Note that if we fold each level (or row) of that isosceles triangle of A237593 we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n).
(End).

Cf. A028982, A067742, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A071562, A259176, A259179, A279387, A286001, A322141, A338204, A338758.

a[n_] := Sum[(-1)^(k + 1) Ceiling[(n + 1)/k - (k + 1)/2], {k, 1, Floor[-1/2 + 1/2 Sqrt[8 n + 1]]}]; Table[a[n], {n, 40}]

a(n) = sum(k=1, floor(-1/2 + 1/2*sqrt(8*n + 1)), (-1)^(k + 1)*ceil((n + 1)/k - (k + 1)/2)); \\ Indranil Ghosh, Apr 21 2017

from sympy import sqrt
import math
def a(n): return sum((-1)**(k + 1) * int(math.ceil((n + 1)/k - (k + 1)/2)) for k in range(1, int(math.floor(-1/2 + 1/2*sqrt(8*n + 1))) + 1))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 21 2017

a(n) = Sum_{k = 1..A003056(n)} (-1)^(k+1) A235791(n,k).
a(n) = A285901(n) - A285902(n), assuming the conjecture 3. - Omar E. Pol, Feb 15 2018
a(n) = n - A322141(n). - Omar E. Pol, Dec 22 2020

More terms from Omar E. Pol, Apr 16 2014

Definition edited by N. J. A. Sloane, Dec 20 2020

A338204 a(n) is the sum of odd-indexed terms (of every row) of the first n rows of the triangle A237591.

Original entry on oeis.org

1, 3, 5, 8, 11, 16, 21, 27, 34, 41, 48, 57, 66, 75, 86, 98, 110, 123, 136, 151, 166, 181, 196, 213, 231, 249, 267, 287, 307, 329, 351, 374, 397, 420, 445, 471, 497, 523, 549, 577, 605, 635, 665, 695, 727, 759, 791, 825, 860, 896, 932, 968, 1004, 1042, 1080, 1120, 1160, 1200, 1240, 1282

Offset: 1

Omar E. Pol, Dec 21 2020

nonn

			Illustration of a(16) = 98 in two ways:
.
Level                                 _              _
1                                   _|_|            |_|_
2                                 _|_ _|            |_ _|
3                               _|_ _|              |_ _|_
4                             _|_ _ _|              |_ _ _|
5                           _|_ _ _|  _             |_ _ _|_ _
6                         _|_ _ _ _| |_|            |_ _ _ _|_|
7                       _|_ _ _ _|   |_|            |_ _ _ _|_|_
8                     _|_ _ _ _ _|  _|_|            |_ _ _ _ _|_|_
9                   _|_ _ _ _ _|   |_ _|            |_ _ _ _ _|_ _|
10                _|_ _ _ _ _ _|   |_|              |_ _ _ _ _ _|_|
11              _|_ _ _ _ _ _|    _|_|              |_ _ _ _ _ _|_|_ _
12            _|_ _ _ _ _ _ _|   |_ _|              |_ _ _ _ _ _ _|_ _|
13          _|_ _ _ _ _ _ _|     |_ _|              |_ _ _ _ _ _ _|_ _|
14        _|_ _ _ _ _ _ _ _|    _|_|  _             |_ _ _ _ _ _ _ _|_|_ _
15      _|_ _ _ _ _ _ _ _|     |_ _| |_|            |_ _ _ _ _ _ _ _|_ _|_|_
16     |_ _ _ _ _ _ _ _ _|     |_ _| |_|            |_ _ _ _ _ _ _ _ _|_ _|_|
...
                  Figure 1.                                 Figure 2.
.
For n = 16, figure 1 shows the illustration of a(16) taken from the isosceles triangle of A237593. There are 98 cells in the first 16 rows of the diagram, so a(16) = 98.
Figure 2 shows the illustration of a(16) taken from an octant of the pyramid described in A244050 and A245092. There are 98 cells in the first 16 rows of the diagram, so a(16) = 98.

Michel Marcus, Table of n, a(n) for n = 1..5000

Partial sums of A240542.

Cf. A000217, A067742, A235791, A237591, A237593, A259176, A286001, A322141, A338758.

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
row237591(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); }
a003056(n) = floor((sqrt(1+8*n)-1)/2);
f(n) = my(row=row237591(n)); sum(k=1, a003056(n), if ((k%2), row[k])); \\ A240542
a(n) = sum(k=1, n, f(k)); \\ Michel Marcus, Dec 22 2020

a(n) = A000217(n) - A338758(n).

A338758 a(n) is the sum of even-indexed terms (of every row) of first n rows of the triangle A237591.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 9, 11, 14, 18, 21, 25, 30, 34, 38, 43, 48, 54, 59, 65, 72, 80, 87, 94, 102, 111, 119, 128, 136, 145, 154, 164, 175, 185, 195, 206, 218, 231, 243, 256, 268, 281, 295, 308, 322, 337, 351, 365, 379, 394, 410, 427, 443, 460, 476, 493, 511, 530, 548, 567, 587, 606, 625

Offset: 1

Omar E. Pol, Dec 21 2020

nonn

			Illustration of a(16) = 38 in two ways:
.
Level
1
2                            _                                      _
3                           |_|                                   _|_|
4                          _|_|                                 _|_|
5                         |_ _|                               _|_ _|
6                        _|_|                               _|_|
7                       |_ _|                             _|_ _|
8                      _|_ _|                           _|_ _|
9                     |_ _|  _                        _|_ _|
10                   _|_ _| |_|                     _|_ _|_|
11                  |_ _ _| |_|                   _|_ _ _|_|
12                 _|_ _|   |_|                 _|_ _|_|
13                |_ _ _|  _|_|               _|_ _ _|_|
14               _|_ _ _| |_ _|             _|_ _ _|_ _|
15              |_ _ _|   |_|             _|_ _ _|_|
16              |_ _ _|   |_|            |_ _ _|_|
...
                    Figure 1.                       Figure 2.
.
For n = 16, figure 1 shows the illustration of a(16) taken from the isosceles triangle of A237593. There are 38 cells in the first 16 levels of the diagram, so a(16) = 38.
Figure 2 shows the illustration of a(16) taken from an octant of the pyramid described in A244050 and A245092. There are 38 cells in the first 16 levels of the diagram, so a(16) = 38.

Partial sums of A322141.

Cf. A000217, A067742, A237591, A237593, A240542, A259177, A286001, A338204.

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
row237591(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); }
a003056(n) = floor((sqrt(1+8*n)-1)/2);
a322141(n) = my(row=row237591(n)); sum(k=1, a003056(n), if (!(k%2), row[k]));
a(n) = sum(k=1, n, a322141(k)); \\ Michel Marcus, Dec 22 2020

a(n) = A000217(n) - A338204(n).

A380231 Alternating row sums of triangle A237591.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 3, 4, 5, 4, 3, 6, 5, 4, 7, 8, 7, 8, 7, 10, 9, 8, 7, 10, 11, 10, 9, 12, 11, 14, 13, 14, 13, 12, 15, 16, 15, 14, 13, 16, 15, 18, 17, 16, 19, 18, 17, 20, 21, 22, 21, 20, 19, 22, 21, 24, 23, 22, 21, 24, 23, 22, 25, 26, 25, 28, 27, 26, 25, 28, 27, 32, 31, 30, 29, 28, 31, 30, 29

Offset: 1

Omar E. Pol, Jan 17 2025

nonn

Consider the symmetric Dyck path in the first quadrant of the square grid described in the n-th row of A237593. Let C = (A240542(n), A240542(n)) be the middle point of the Dyck path.
a(n) is also the coordinate on the x axis of the point (a(n),n) and also the coordinate on the y axis of the point (n,a(n)) such that the middle point of the line segment [(a(n),n),(n,a(n))] coincides with the middle point C of the symmetric Dyck path.
The three line segments [(a(n),n),C], [(n,a(n)),C] and [(n,n),C] have the same length.
For n > 2 the points (n,n), C and (a(n),n) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (n,n), C and (n,a(n)) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (a(n),n), (n,n) and (n,a(n)) are the vertices of a virtual isosceles right triangle.

			For n = 14 the 14th row of A237591 is [8, 3, 1, 2] hence the alternating row sum is 8 - 3 + 1 - 2 = 4, so a(14) = 4.
On the other hand the 14th row of A237593 is the 14th row of A237591 together with the 14 th row of A237591 in reverse order as follows: [8, 3, 1, 2, 2, 1, 3, 8].
Then with the terms of the 14th row of A237593 we can draw a Dyck path in the first quadrant of the square grid as shown below:
.
         (y axis)
          .
          .
          .    (4,14)              (14,14)
          ._ _ _ . _ _ _ _            .
          .               |
          .               |
          .               |_
          .                 |
          .                 |_ _
          .                C    |_ _ _
          .                           |
          .                           |
          .                           |
          .                           |
          .                           . (14,4)
          .                           |
          .                           |
          . . . . . . . . . . . . . . | . . . (x axis)
        (0,0)
.
In the example the point C is the point (9,9).
The three line segments [(4,14),(9,9)], [(14,4),(9,9)] and [(14,14),(9,9)] have the same length.
The points (14,14), (9,9) and (4,14) are the vertices of a virtual isosceles right triangle.
The points (14,14), (9,9) and (14,4) are the vertices of a virtual isosceles right triangle.
The points (4,14), (14,14) and (14,4) are the vertices of a virtual isosceles right triangle.

Other alternating row sums (ARS) related to the Dyck paths of A237593 and the stepped pyramid described in A245092 are as follows:
ARS of A237593 give A000004.
ARS of A196020 give A000203.
ARS of A252117 give A000203.
ARS of A271343 give A000593.
ARS of A231347 give A001065.
ARS of A236112 give A004125.
ARS of A236104 give A024916.
ARS of A249120 give A024916.
ARS of A271344 give A033879.
ARS of A231345 give A033880.
ARS of A239313 give A048050.
ARS of A237048 give A067742.
ARS of A236106 give A074400.
ARS of A235794 give A120444.
ARS of A266537 give A146076.
ARS of A236540 give A153485.
ARS of A262612 give A175254.
ARS of A353690 give A175254.
ARS of A239446 give A235796.
ARS of A239662 give A239050.
ARS of A235791 give A240542.
ARS of A272026 give A272027.
ARS of A211343 give A336305.
Cf. A221529, A237270, A237271, A237591, A262626, A322141.

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
a(n) = my(orow = concat(row235791(n), 0)); vecsum(vector(#orow-1, i, (-1)^(i+1)*(orow[i] - orow[i+1]))); \\ Michel Marcus, Apr 13 2025

a(n) = 2*A240542(n) - n.
a(n) = n - 2*A322141(n).
a(n) = A240542(n) - A322141(n).

cp's OEIS Frontend

A240542 The midpoint of the (rotated) Dyck path from (0, n) to (n, 0) defined by A237593 has coordinates (a(n), a(n)). Also a(n) is the alternating sum of the n-th row of A235791.

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A338204 a(n) is the sum of odd-indexed terms (of every row) of the first n rows of the triangle A237591.

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A338758 a(n) is the sum of even-indexed terms (of every row) of first n rows of the triangle A237591.

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A380231 Alternating row sums of triangle A237591.

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