A259179 -id:A259179 - cp's OEIS frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

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.

Original entry on oeis.org

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

A279286 a(n) = d if the point (d,d) is shared by a record of different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

Original entry on oeis.org

1, 2, 7, 15, 52, 102, 296, 371, 455, 929, 1853, 2034, 4517, 4797, 5829, 6146, 6948, 17577, 18915, 60349, 78369, 85171, 123788, 128596, 415355, 906771, 1308771, 3329668

Offset: 1

Omar E. Pol, Dec 09 2016

Is this sequence infinite?
First differs from A282197 (another version) at a(19). - Omar E. Pol, Feb 08 2017
a(n) = d if the point (d,d) belongs to a vertical-line-segment whose length is a record in the main diagonal of the pyramid described in A245092 (starting from the top). The diagram of the symmetries of sigma is also the top view of the mentioned pyramid. See examples. - Omar E. Pol, Feb 09 2017

			The first record of height difference is between the levels 1 and 2 of the pyramid (starting from the top), at the point (1,1) of the main diagonal of the top view of the pyramid, so a(1) = 1.
The second record of height difference is between the levels 2 and 4, at the point (2,2) of the main diagonal of the top view of the pyramid, so a(2) = 2.
The third record of height difference is between the levels 9 and 12, at the point (7,7) of the main diagonal of the top view of the pyramid, so a(3) = 7.
The fourth record of height difference is between the levels 20 and 24, at the point (15,15) of the main diagonal of the top view of the pyramid, so a(4) = 15.
Illustration of the diagram of the symmetries of sigma (n = 1..16), which is also the top view of the pyramid described in A245092, and it is also a quadrant of the top view of the pyramid described in A244050:
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.    |_| | | | | | | | | | | | | | | |
.    |_ _|_| | | | | | | | | | | | | |
.    |_ _|  _|_| | | | | | | | | | | |
.    |_ _ _|    _|_| | | | | | | | | |
.    |_ _ _|  _|  _ _|_| | | | | | | |
.    |_ _ _ _|  _| |  _ _|_| | | | | |
.    |_ _ _ _| |_ _|_|    _ _|_| | | |
.    |_ _ _ _ _|  _|     |  _ _ _|_| |
.    |_ _ _ _ _| |      _|_| |  _ _ _|
.    |_ _ _ _ _ _|  _ _|    _| |
.    |_ _ _ _ _ _| |  _|  _|  _|
.    |_ _ _ _ _ _ _| |_ _|  _|
.    |_ _ _ _ _ _ _| |  _ _|
.    |_ _ _ _ _ _ _ _| |
.    |_ _ _ _ _ _ _ _| |
.    |_ _ _ _ _ _ _ _ _|
...

Where records occur in A259179, (was the original Name).

Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A240542, A245092, A244050, A262626, A282197.

a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a279286[b_] := Module[{centers={{1, 1}}, acc={1}, k=2, cPrev=1, cCur, len}, While[k<=b, cCur=a240542[k]; If[Last[acc]==cCur, AppendTo[acc, cCur], len=Length[acc]; If[First[Last[centers]]Hartmut F. W. Hoft, Feb 08 2017 *)

a(7)-a(28) from Hartmut F. W. Hoft, Feb 08 2017

New Name from Omar E. Pol, Feb 09 2017

A259176 Triangle read by rows T(n,k) in which row n lists the odd-indexed terms of n-th row of triangle A237593.

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 3, 2, 4, 1, 1, 4, 1, 2, 5, 1, 2, 5, 2, 2, 6, 1, 1, 2, 6, 1, 1, 3, 7, 2, 1, 2, 7, 2, 1, 3, 8, 1, 2, 3, 8, 2, 1, 1, 3, 9, 2, 1, 1, 3, 9, 2, 1, 1, 4, 10, 2, 1, 2, 3, 10, 2, 1, 2, 4, 11, 2, 2, 1, 4, 11, 3, 1, 1, 1, 4, 12, 2, 1, 1, 2, 4, 12, 2, 1, 1, 2, 5, 13, 3, 1, 1, 2, 4, 13, 3, 2, 1, 1, 5, 14, 2, 2, 1, 2, 5

Offset: 1

Omar E. Pol, Aug 15 2015

Row n has length A003056(n) hence column k starts in row A000217(k).

Row n is a permutation of the n-th row of A237591 for some n, hence the sequence is a permutation of A237591.

			Written as an irregular triangle the sequence begins:
1;
2;
2, 1;
3, 1;
3, 2;
4, 1, 1;
4, 1, 2;
5, 1, 2;
5, 2, 2;
6, 1, 1, 2;
6, 1, 1, 3;
7, 2, 1, 2;
7, 2, 1, 3;
8, 1, 2, 3;
8, 2, 1, 1, 3;
9, 2, 1, 1, 3;
...
Illustration of initial terms (side view of the pyramid):
Row   _
1    |_|_
2    |_ _|_
3    |_ _|_|_
4    |_ _ _|_|_
5    |_ _ _|_ _|_
6    |_ _ _ _|_|_|_
7    |_ _ _ _|_|_ _|_
8    |_ _ _ _ _|_|_ _|_
9    |_ _ _ _ _|_ _|_ _|_
10   |_ _ _ _ _ _|_|_|_ _|_
11   |_ _ _ _ _ _|_|_|_ _ _|_
12   |_ _ _ _ _ _ _|_ _|_|_ _|_
13   |_ _ _ _ _ _ _|_ _|_|_ _ _|_
14   |_ _ _ _ _ _ _ _|_|_ _|_ _ _|_
15   |_ _ _ _ _ _ _ _|_ _|_|_|_ _ _|_
16   |_ _ _ _ _ _ _ _ _|_ _|_|_|_ _ _|
...
The above structure represents the first 16 levels (starting from the top) of one of the side views of the infinite stepped pyramid described in A245092. For another side view see A259177.
.
Illustration of initial terms (partial front view of the pyramid):
Row                                 _
1                                 _|_|
2                               _|_ _|_
3                             _|_ _| |_|
4                           _|_ _ _| |_|_
5                         _|_ _ _|  _|_ _|
6                       _|_ _ _ _| |_| |_|_
7                     _|_ _ _ _|   |_| |_ _|
8                   _|_ _ _ _ _|  _|_| |_ _|_
9                 _|_ _ _ _ _|   |_ _|_  |_ _|
10              _|_ _ _ _ _ _|   |_| |_| |_ _|_
11            _|_ _ _ _ _ _|    _|_| |_| |_ _ _|
12          _|_ _ _ _ _ _ _|   |_ _| |_|   |_ _|_
13        _|_ _ _ _ _ _ _|     |_ _| |_|_  |_ _ _|
14      _|_ _ _ _ _ _ _ _|    _|_|  _|_ _| |_ _ _|_
15    _|_ _ _ _ _ _ _ _|     |_ _| |_| |_|   |_ _ _|
16   |_ _ _ _ _ _ _ _ _|     |_ _| |_| |_|   |_ _ _|
...
A part of the hidden pattern of the symmetric representation of sigma emerges from the partial front view of the pyramid described in A245092.
For another partial front view see A259177. For the total front view see A237593.

Bisection of A237593.
Row sums give A000027.
For the mirror see A259177 which is another bisection of A237593.
Cf. A000203, A000217, A003056, A024916, A175254, A196020, A236104, A237270, A237271, A237591, A244580, A245092, A249351, A259179, A261350.

(* function f[n,k] and its support functions are defined in A237593 *)
a259176[n_, k_] := f[n, 2*k-1]
TableForm[Table[a259176[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a259176[n, k], {n, 1, 26}, {k, 1, [n]}]] (* sequence data *)
(* Hartmut F. W. Hoft, Mar 06 2017 *)

row(n) = (sqrt(8*n + 1) - 1)\2;
s(n, k) = ceil((n + 1)/k - (k + 1)/2) - ceil((n + 1)/(k + 1) - (k + 2)/2);
T(n, k) = if(k<=row(n), s(n, k), s(n, 2*row(n) + 1 - k));
a259177(n, k) = T(n, 2*k - 1);
for(n=1, 26, for(k=1, row(n), print1(a259177(n, k),", ");); print();)  \\ Indranil Ghosh, Apr 21 2017

from sympy import sqrt
import math
def row(n): return int(math.floor((sqrt(8*n + 1) - 1)/2))
def s(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2)) - int(math.ceil((n + 1)/(k + 1) - (k + 2)/2))
def T(n, k): return s(n, k) if k<=row(n) else s(n, 2*row(n) + 1 - k)
def a259177(n, k): return T(n, 2*k - 1)
for n in range(1, 11): print([a259177(n, k) for k in range(1, row(n) + 1)]) # Indranil Ghosh, Apr 21 2017

Better definition from Omar E. Pol, Apr 26 2021

A259177 Triangle read by rows T(n,k) in which row n lists the even-indexed terms of n-th row of triangle A237593.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 3, 1, 1, 4, 2, 1, 4, 2, 1, 5, 2, 2, 5, 2, 1, 1, 6, 3, 1, 1, 6, 2, 1, 2, 7, 3, 1, 2, 7, 3, 2, 1, 8, 3, 1, 1, 2, 8, 3, 1, 1, 2, 9, 4, 1, 1, 2, 9, 3, 2, 1, 2, 10, 4, 2, 1, 2, 10, 4, 1, 2, 2, 11, 4, 1, 1, 1, 3, 11, 4, 2, 1, 1, 2, 12, 5, 2, 1, 1, 2, 12, 4, 2, 1, 1, 3, 13, 5, 1, 1, 2, 3, 13, 5, 2, 1, 2, 2, 14

Offset: 1

Omar E. Pol, Aug 15 2015

Row n has length A003056(n) hence column k starts in row A000217(k).

Row n is a permutation of the n-th row of A237591 for some n, hence the sequence is a permutation of A237591.

			Written as an irregular triangle the sequence begins:
1;
2;
1, 2;
1, 3;
2, 3;
1, 1, 4;
2, 1, 4;
2, 1, 5;
2, 2, 5;
2, 1, 1, 6;
3, 1, 1, 6;
2, 1, 2, 7;
3, 1, 2, 7;
3, 2, 1, 8;
3, 1, 1, 2, 8;
3, 1, 1, 2, 9;
...
Illustration of initial terms (side view of the pyramid):
Row                                 _
1                                 _|_|
2                               _|_ _|
3                             _|_|_ _|
4                           _|_|_ _ _|
5                         _|_ _|_ _ _|
6                       _|_|_|_ _ _ _|
7                     _|_ _|_|_ _ _ _|
8                   _|_ _|_|_ _ _ _ _|
9                 _|_ _|_ _|_ _ _ _ _|
10              _|_ _|_|_|_ _ _ _ _ _|
11            _|_ _ _|_|_|_ _ _ _ _ _|
12          _|_ _|_|_ _|_ _ _ _ _ _ _|
13        _|_ _ _|_|_ _|_ _ _ _ _ _ _|
14      _|_ _ _|_ _|_|_ _ _ _ _ _ _ _|
15    _|_ _ _|_|_|_ _|_ _ _ _ _ _ _ _|
16   |_ _ _|_|_|_ _|_ _ _ _ _ _ _ _ _|
...
The above structure represents the first 16 levels (starting from the top) of one of the side views of the infinite stepped pyramid described in A245092. For another side view see A259176.
.
Illustration of initial terms (partial front view of the pyramid):
Row                                 _
1                                  |_|_
2                                 _|_ _|_
3                                |_| |_ _|_
4                               _|_| |_ _ _|_
5                              |_ _|_  |_ _ _|_
6                             _|_| |_| |_ _ _ _|_
7                            |_ _| |_|   |_ _ _ _|_
8                           _|_ _| |_|_  |_ _ _ _ _|_
9                          |_ _|  _|_ _|   |_ _ _ _ _|_
10                        _|_ _| |_| |_|   |_ _ _ _ _ _|_
11                       |_ _ _| |_| |_|_    |_ _ _ _ _ _|_
12                      _|_ _|   |_| |_ _|   |_ _ _ _ _ _ _|_
13                     |_ _ _|  _|_| |_ _|     |_ _ _ _ _ _ _|_
14                    _|_ _ _| |_ _|_  |_|_    |_ _ _ _ _ _ _ _|_
15                   |_ _ _|   |_| |_| |_ _|     |_ _ _ _ _ _ _ _|_
16                   |_ _ _|   |_| |_| |_ _|     |_ _ _ _ _ _ _ _ _|
...
A part of the hidden pattern of the symmetric representation of sigma emerges from the partial front view of the pyramid described in A245092.
For another partial front view see A259176. For the total front view see A237593.

Bisection of A237593.
Row sums give A000027.
Mirror of A259176 which is another bisection of A237593.
Cf. A000203, A000217, A003056, A024916, A175254, A196020, A236104, A237270, A237271, A237591, A244580, A245092, A249351, A259179, A261350.

(* function f[n,k] and its support functions are defined in A237593 *)
a259177[n_, k_] := f[n, 2*k]
TableForm[Table[a259177[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a259177[n, k], {n, 1, 26}, {k, 1, [n]}]] (* sequence data *)
(* Hartmut F. W. Hoft, Mar 06 2017 *)

row(n) = (sqrt(8*n + 1) - 1)\2;
s(n, k) = ceil((n + 1)/k - (k + 1)/2) - ceil((n + 1)/(k + 1) - (k + 2)/2);
T(n, k) = if(k<=row(n), s(n, k), s(n, 2*row(n) + 1 - k));
a259177(n, k) = T(n, 2*k);
for(n=1, 26, for(k=1, row(n), print1(a259177(n, k),", ");); print();) \\ Indranil Ghosh, Apr 21 2017

from sympy import sqrt
import math
def row(n): return int(math.floor((sqrt(8*n + 1) - 1)/2))
def s(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2)) - int(math.ceil((n + 1)/(k + 1) - (k + 2)/2))
def T(n, k): return s(n, k) if k<=row(n) else s(n, 2*row(n) + 1 - k)
def a259177(n, k): return T(n, 2*k)
for n in range(1, 27): print([a259177(n, k) for k in range(1, row(n) + 1)]) # Indranil Ghosh, Apr 21 2017

Better definition from Omar E. Pol, Apr 26 2021

A280919 Precipices from the successive terraces, descending by the main diagonal of the pyramid described in A245092. Also first differences of A071562.

Original entry on oeis.org

1, 2, 2, 2, 1, 3, 3, 1, 2, 2, 4, 1, 3, 2, 2, 3, 1, 4, 2, 3, 3, 1, 1, 4, 2, 4, 3, 1, 2, 4, 2, 5, 3, 1, 3, 4, 2, 1, 5, 2, 1, 1, 4, 4, 2, 2, 5, 3, 1, 5, 2, 2, 2, 3, 5, 3, 1, 6, 3, 1, 2, 4, 2, 3, 3, 1, 1, 6, 4, 2, 5, 3, 2, 3, 1, 2, 2, 4, 4, 1, 1, 6, 4, 1, 3, 1, 3

Offset: 1

Omar E. Pol, Jan 10 2017

nonn

Descending by the main diagonal of the pyramid, A071562 gives the levels where we can find a terrace.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k).
a(n) is the length of the n-th vertical line segment at the main diagonal of the pyramid.
a(n) is the precipice of A071562(n).
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
Equals nonzero terms of A259179. - Omar E. Pol, Apr 17 2018

Robert Price, Table of n, a(n) for n = 1..13750
Omar E. Pol, Illustration of the isosceles triangle of A237593 (rows 1..28)
Omar E. Pol, Perspective view of the pyramid (first 16 levels)

For more information about the precipices see A276112, A277437, A280223 and A280295.

Cf. A000203, A071562, A196020, A235791, A236104, A237048, A237591, A237593, A237270, A237271, A244050, A245092, A259179, A262626.

Differences@ Select[Range@ 228, Function[n, Total@ Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] != 0]] (* Michael De Vlieger, Jan 13 2017, after Robert G. Wilson v at A071562 *)

a(n) = A280223(A071562(n)).

More terms from Michael De Vlieger, Jan 13 2017

A261350 Triangle read by rows T(n,k) which is the mirror of A237591.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 3, 1, 1, 4, 1, 2, 4, 1, 2, 5, 2, 2, 5, 1, 1, 2, 6, 1, 1, 3, 6, 1, 2, 2, 7, 1, 2, 3, 7, 2, 1, 3, 8, 1, 1, 2, 3, 8, 1, 1, 2, 3, 9, 1, 1, 2, 4, 9, 1, 2, 2, 3, 10, 1, 2, 2, 4, 10, 2, 1, 2, 4, 11, 1, 1, 1, 3, 4, 11, 1, 1, 2, 2, 4, 12, 1, 1, 2, 2, 5, 12, 1, 1, 2, 3, 4, 13, 1, 2, 1, 3, 5, 13, 1, 2, 2, 2, 5, 14

Offset: 1

Omar E. Pol, Aug 18 2015

Row n has length A003056(n) hence column k starts in row A000217(k).
Row sums give A000027.
Right border gives A008619, n >= 1.
n is an odd prime if and only if T(n,r-1) = 1 + T(n-1,r-1) and T(n,k) = T(n-1,k) for the rest of the values of k, where r = A003056(n) is the number of elements in row n.
T(n,k) is also the length of the k-th segment in a zig-zag path on the first quadrant of the square grid, connecting the point (m, m) with the point (0, n), ending with a segment in horizontal direction, where m = A240542(n). The area of the polygon defined by the y-axis, this zig-zag path and the diagonal [(0, 0), (m, m)], is equal to A024916(n)/2, one half of the sum of all divisors of all positive integers <= n. Therefore the reflected polygon, which is adjacent to the x-axis, with the zig-zag path connecting the point (n, 0) with the point (m, m), has the same property. And so on for each octant in the four quadrants.
For the representation of A024916 and A000203 we use two octants, for example: the first octant and the second octant, or the 6th octant and the 7th octant, etc., see A237593.
The elements of the n-th row of A237591 together with the elements of the n-th row of this sequence give the n-th row of A237593.
The connection between A196020 and A237271 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> this sequence --> A237593 --> A239660 --> A237270 --> A237271.
T(n,k) is also the area (or the number of cells) of the k-th vertical side at the n-th level (starting from the top) in the right part of the front view of the stepped pyramid described in A245092, see Example section.

			Triangle begins:
Row
1                       1;
2                       2;
3                    1, 2;
4                    1, 3;
5                    2, 3;
6                 1, 1, 4;
7                 1, 2, 4;
8                 1, 2, 5;
9                 2, 2, 5;
10             1, 1, 2, 6;
11             1, 1, 3, 6;
12             1, 2, 2, 7;
13             1, 2, 3, 7;
14             2, 1, 3, 8;
15          1, 1, 2, 3, 8;
16          1, 1, 2, 3, 9;
17          1, 1, 2, 4, 9;
18          1, 2, 2, 3, 10;
19          1, 2, 2, 4, 10;
20          2, 1, 2, 4, 11;
21       1, 1, 1, 3, 4, 11;
22       1, 1, 2, 2, 4, 12;
23       1, 1, 2, 2, 5, 12;
24       1, 1, 2, 3, 4, 13;
25       1, 2, 1, 3, 5, 13;
26       1, 2, 2, 2, 5, 14;
...
Illustration of initial terms:
Row      _
1       |1|_
2       |_ 2|_
3       |1|  2|_
4       |1|_   3|_
5       |_ 2|    3|_
6       |1|1|_     4|_
7       |1|  2|      4|_
8       |1|_ 2|_       5|_
9       |_ 2|  2|        5|_
10      |1|1|  2|_         6|_
11      |1|1|_   3|          6|_
12      |1|  2|  2|_           7|_
13      |1|_ 2|    3|            7|_
14      |_ 2|1|_   3|_             8|_
15      |1|1|  2|    3|              8|_
16      |1|1|  2|    3|_               9|_
17      |1|1|_ 2|_     4|                9|_
18      |1|  2|  2|    3|_                10|_
19      |1|_ 2|  2|      4|                 10|_
20      |_ 2|1|  2|_     4|_                  11|_
21      |1|1|1|_   3|      4|                   11|_
22      |1|1|  2|  2|      4|_                    12|_
23      |1|1|  2|  2|_       5|                     12|_
24      |1|1|_ 2|    3|      4|_                      13|_
25      |1|  2|1|_   3|        5|                       13|_
26      |1|  2|  2|  2|        5|                         14|
...
Also the diagram represents the right part of the front view of the pyramid described in A245092. For the other half front view see A237591. For more information about the pyramid and the symmetric representation of sigma see A237593.

Cf. A000203, A000217, A003056, A008016, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244580, A245092, A249351, A259176, A259177, A259179.

A279385 Irregular triangle read by rows in which row n lists the numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n), or row n is 0 if no such k exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 0, 6, 7, 8, 9, 10, 11, 0, 12, 13, 14, 0, 15, 16, 17, 18, 19, 0, 20, 21, 22, 23, 0, 24, 25, 26, 27, 0, 28, 29, 0, 30, 31, 32, 33, 34, 0, 35, 36, 37, 38, 39, 0, 40, 41, 0, 42, 43, 44, 0, 45, 46, 47, 0, 48, 49, 50, 51, 52, 53, 0, 54, 55, 0, 56, 57, 58, 59, 0, 60, 61, 62, 0, 63, 64, 65, 0, 66, 67, 68, 69, 0

Offset: 1

Omar E. Pol, Dec 12 2016

For more information about the mentioned Dyck paths see A237593.

			n         Triangle begins:
1         1;
2         2, 3;
3         4, 5;
4         0;
5         6, 7;
6         8,
7         9, 10, 11;
8         0;
9         12, 13, 14;
10        0;
11        15;
12        16, 17;
13        18, 19;
14        0;
15        20, 21, 22, 23;
16        0;
...

Positive terms give A000027.
Cf. A259179(n) is the number of positive terms in row n.
Cf. A000203, A196020, A236104, A235791, A237048, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286.

(* last computed value is dropped to avoid a potential under count of crossings *)
a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]
pathGroups[n_] := Module[{t}, t=Table[{}, a240542[n]]; Map[AppendTo[t[[a240542[#]]], #]&, Range[n]]; Map[If[t[[#]]=={}, t[[#]]={0}]&, Range[Length[t]]]; Most[t]]
a279385[n_] := Flatten[pathGroups[n]]
a279385[70] (* sequence *)
a279385T[n_] := TableForm[pathGroups[n], TableHeadings->{Range[a240542[n]-1], None}]
a279385T[24] (* display of irregular triangle - Hartmut F. W. Hoft, Feb 02 2022 *)

More terms from Omar E. Pol, Jun 20 2018

A280223 Precipice of n: descending by the main diagonal of the pyramid described in A245092, a(n) is the height difference between the n-th level (starting from the top) and the level of the next terrace.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 1, 4, 3, 2, 1, 2, 1, 4, 3, 2, 1, 3, 2, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 5, 4, 3, 2, 1, 3, 2, 1, 1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 1, 5, 4, 3, 2, 1, 2, 1, 1, 1, 4, 3, 2, 1, 4, 3, 2, 1

Offset: 1

Omar E. Pol, Dec 29 2016

nonn

The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the n-th level of the pyramid are also the parts of the symmetric representation of sigma(n).
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
Note that if a(n) > 1 then the next k terms are the first k positive integers in decreasing order, where k = a(n) - 1.
For more information about the precipices see A277437 and A280295.
a(n) is also the number of numbers >= n whose largest Dyck paths of the symmetric representation of sigma share the same point at the main diagonal of the diagram. For more information see A237593.

			Descending by the main diagonal of the stepped pyramid, for the levels 9, 10 and 11 we have that the next terrace is in the 12th level, so a(9) = 12 - 9 = 3, a(10) = 12 - 10 = 2, and a(11) = 12 - 11 = 1.

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A277437, A279286, A279385, A280295.

More terms from Omar E. Pol, Jan 02 2017

A277437 Square array read by antidiagonals upwards in which T(n,k) is the n-th number j such that, descending by the main diagonal of the pyramid described in A245092, the height difference between the level j (starting from the top) and the level of the next terrace is equal to k.

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 7, 6, 12, 20, 8, 10, 21, 36, 72, 11, 13, 25, 50, 91, 144, 14, 16, 32, 56, 112

Offset: 1

Omar E. Pol, Dec 29 2016

This is a permutation of the natural numbers.
Column k lists the numbers with precipice k. For more information about the precipices see A280223 and A280295.
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
If a number m is in the column k and k > 1 then m + 1 is the column k - 1.
The largest Dyck path of the symmetric representations of next k - 1 positive integers greater than T(n,k) shares the middle point of the largest Dyck path of the symmetric representation of sigma(T(n,k)). For more information see A237593.

			The corner of the square array begins:
   1,  2,  9, 20, 72, 144,
   3,  4, 12, 36, 91,
   5,  6, 21, 50,
   7, 10, 25,
   8, 13,
  11,
  ...
T(1,6) = 144 because it is the smallest number with precipice 6.

Row 1 gives A280295.
Column 1 gives A276112.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223.

T(n,1) = A071562(n+1) - 1.

a(20)-a(26) from Omar E. Pol, Jan 02 2017

cp's OEIS Frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

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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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A279286 a(n) = d if the point (d,d) is shared by a record of different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

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A259176 Triangle read by rows T(n,k) in which row n lists the odd-indexed terms of n-th row of triangle A237593.

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A259177 Triangle read by rows T(n,k) in which row n lists the even-indexed terms of n-th row of triangle A237593.

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A280919 Precipices from the successive terraces, descending by the main diagonal of the pyramid described in A245092. Also first differences of A071562.

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A261350 Triangle read by rows T(n,k) which is the mirror of A237591.

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A279385 Irregular triangle read by rows in which row n lists the numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n), or row n is 0 if no such k exists.

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