cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Sep 26 2015

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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.

A239660 Triangle read by rows in which row n lists two copies of the n-th row of triangle A237593.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4, 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4, 5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5, 5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5, 6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6, 6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6
Offset: 1

Views

Author

Omar E. Pol, Mar 24 2014

Keywords

Comments

For the construction of this sequence also we can start from A235791.
This sequence can be interpreted as an infinite Dyck path: UDUDUUDD...
Also we use this sequence for the construction of a spiral in which the arms in the quadrants give the symmetric representation of sigma, see example.
We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016
The spiral has the property that the sum of the parts in the quadrants 1 and 3, divided by the sum of the parts in the quadrants 2 and 4, converges to 3/5. - Omar E. Pol, Jun 10 2019

Examples

			Triangle begins (first 15.5 rows):
1, 1, 1, 1;
2, 2, 2, 2;
2, 1, 1, 2, 2, 1, 1, 2;
3, 1, 1, 3, 3, 1, 1, 3;
3, 2, 2, 3, 3, 2, 2, 3;
4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4;
4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4;
5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5;
5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5;
6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6;
6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6;
7, 2, 2, 1, 1, 2, 2, 7, 7, 2, 2, 1, 1, 2, 2, 7;
7, 3, 2, 1, 1, 2, 3, 7, 7, 3, 2, 1, 1, 2, 3, 7;
8, 3, 1, 2, 2, 1, 3, 8, 8, 3, 1, 2, 2, 1, 3, 8;
8, 3, 2, 1, 1, 1, 1, 2, 3, 8, 8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
9, 3, 2, 1, 1, 1, 1, 2, 3, 9, ...
.
Illustration of initial terms as an infinite Dyck path (row n = 1..4):
.
.                            /\/\    /\/\
.       /\  /\  /\/\  /\/\  /    \  /    \
.  /\/\/  \/  \/    \/    \/      \/      \
.
.
Illustration of initial terms for the construction of a spiral related to sigma:
.
.  row 1     row 2          row 3           row 4
.                                          _ _ _
.                                               |_
.             _ _                                 |
.   _ _      |                                    |
.  |   |     |                                    |
.            |         |           |              |
.            |_ _      |_         _|              |
.                        |_ _ _ _|               _|
.                                          _ _ _|
.
.[1,1,1,1] [2,2,2,2] [2,1,1,2,2,1,1,2] [3,1,1,3,3,1,1,3]
.
The first 2*A003056(n) terms of the n-th row are represented in the A010883(n-1) quadrant and the last 2*A003056(n) terms of the n-th row are represented in the A010883(n) quadrant.
.
Illustration of the spiral constructed with the first 15.5 rows of triangle:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.                _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_
.      _ _ _| |      _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |      _ _| |   12 _ _ _ _          |_  |         | |
.     | |     |  _ _|    _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |      _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|    _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|     | |     | |
.   | |     |_|_ _        |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |    _|    _ _|     | |
.   |_|_ _ _        |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _      |                     15|      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|    _| |
.        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |      _|  _|
.             |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|
.                 |                             28|  _ _|
.                 |_ _ _ _ _ _ _ _                | |
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains A237590(16) = 27 parts.
The total area (also the total number of cells) in the n-th arm of the spiral is equal to sigma(n) = A000203(n), considering every quadrant and the axes x and y. (checked by hand up to row n = 128). The parts of the spiral are in A237270: 1, 3, 2, 2, 7...
Diagram extended by _Omar E. Pol_, Aug 23 2018

Links

Crossrefs

Row n has length 4*A003056(n).
The sum of row n is equal to 4*n = A008586(n).
Row n is a palindromic composition of 4*n = A008586(n).
Both column 1 and right border are A008619, n >= 1.
The connection between A196020 and A237270 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> this sequence --> A237270.
Cf. A000203, A000217, A003056, A008619, A010883, A112610, A193553, A237048, A237271, A237590, A239052, A239053, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A262626, A296508, A299778.

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

Views

Author

Hartmut F. W. Hoft, Apr 07 2014

Keywords

Comments

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

Examples

			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).

Crossrefs

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

Programs

  • Mathematica
    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}]
  • PARI
    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
  • Python
    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

Formula

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

Extensions

More terms from Omar E. Pol, Apr 16 2014
Definition edited by N. J. A. Sloane, Dec 20 2020

A259179 Number of Dyck paths described in A237593 that contain the point (n,n) in the diagram of the symmetric representation of sigma.

Original entry on oeis.org

1, 2, 2, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 0, 4, 0, 1, 3, 0, 2, 0, 2, 3, 0, 1, 4, 0, 2, 0, 3, 0, 3, 0, 1, 1, 4, 0, 2, 0, 4, 0, 3, 0, 1, 2, 0, 4, 0, 2, 0, 0, 5, 0, 3, 0, 1, 3, 0, 4, 0, 2, 0, 1, 0, 5, 0, 2, 1, 0, 1, 4, 0, 4, 0, 2, 0, 2, 0, 5, 0, 3, 0, 0, 0, 1, 5, 0, 2, 2, 0, 2, 0, 3, 0, 5, 0, 3, 0, 1, 0, 0, 6
Offset: 1

Views

Author

Omar E. Pol, Aug 11 2015

Keywords

Comments

Since the diagram of the symmetric representation of sigma is also the top view of the stepped pyramid described in A245092, and the diagram is also the top view of the staircase described in A244580, so we have that a(n) is also the height difference (or length of the vertical line segment) at the point (n,n) in the main diagonal of the mentioned structures.
a(n) is the number of occurrences of n in A240542. - Omar E. Pol, Dec 09 2016
Nonzero terms give A280919, the first differences of A071562. - Omar E. Pol, Apr 17 2018
Also first differences of A244367. Where records occur gives A279286. - Omar E. Pol, Apr 20 2020

Examples

			Illustration of initial terms:
--------------------------------------------------------
                           Diagram with 15 Dyck paths
n   A000203(n)  a(n)         to evaluate a(1)..a(10)
--------------------------------------------------------
.                         _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1        1        1      |_| | | | | | | | | | | | | | |
2        3        2      |_ _|_| | | | | | | | | | | | |
3        4        2      |_ _|  _|_| | | | | | | | | | |
4        7        0      |_ _ _|    _|_| | | | | | | | |
5        6        2      |_ _ _|  _|  _ _|_| | | | | | |
6       12        1      |_ _ _ _|  _| |  _ _|_| | | | |
7        8        3      |_ _ _ _| |_ _|_|    _ _|_| | |
8       15        0      |_ _ _ _ _|  _|     |  _ _ _|_|
9       13        3      |_ _ _ _ _| |      _|_| |
10      18        0      |_ _ _ _ _ _|  _ _|    _|
.                        |_ _ _ _ _ _| |  _|  _|
.                        |_ _ _ _ _ _ _| |_ _|
.                        |_ _ _ _ _ _ _| |
.                        |_ _ _ _ _ _ _ _|
.                        |_ _ _ _ _ _ _ _|
.
For n = 3 there are two Dyck paths that contain the point (3,3) so a(3) = 2.
For n = 4 there are no Dyck paths that contain the point (4,4) so a(4) = 0.

Crossrefs

Cf. A000203, A024916, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A240542, A244050, A244367, A244580, A245092, A249351, A256533, A259179, A262626, A279286, A280919.

Programs

  • Mathematica
    a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
a259179[n_] := Module[{t=Table[0, n], k=1, d=1}, While[d<=n, t[[d]]+=1; d=a240542[++k]]; t] (* a(1..n) *)
a259179[102] (* Hartmut F. W. Hoft, Aug 06 2020 *)

Extensions

More terms from Omar E. Pol, Dec 09 2016

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

Views

Author

Omar E. Pol, Dec 09 2016

Keywords

Comments

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

Examples

			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:
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.    |_| | | | | | | | | | | | | | | |
.    |_ _|_| | | | | | | | | | | | | |
.    |_ _|  _|_| | | | | | | | | | | |
.    |_ _ _|    _|_| | | | | | | | | |
.    |_ _ _|  _|  _ _|_| | | | | | | |
.    |_ _ _ _|  _| |  _ _|_| | | | | |
.    |_ _ _ _| |_ _|_|    _ _|_| | | |
.    |_ _ _ _ _|  _|     |  _ _ _|_| |
.    |_ _ _ _ _| |      _|_| |  _ _ _|
.    |_ _ _ _ _ _|  _ _|    _| |
.    |_ _ _ _ _ _| |  _|  _|  _|
.    |_ _ _ _ _ _ _| |_ _|  _|
.    |_ _ _ _ _ _ _| |  _ _|
.    |_ _ _ _ _ _ _ _| |
.    |_ _ _ _ _ _ _ _| |
.    |_ _ _ _ _ _ _ _ _|
...

Crossrefs

Where records occur in A259179, (was the original Name).
Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A240542, A245092, A244050, A262626, A282197.

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Omar E. Pol, Aug 15 2015

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    (* 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 *)
  • PARI
    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
  • Python
    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

Extensions

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

Views

Author

Omar E. Pol, Aug 15 2015

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    (* 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 *)
  • PARI
    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
  • Python
    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

Extensions

Better definition from Omar E. Pol, Apr 26 2021

A262048 Irregular triangle read by rows: T(n, k) = A262045(n, k) * A237593(n, k), n >= 1, 1 <= k <= 2 * A003056(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 0, 0, 2, 3, 1, 1, 3, 3, 0, 0, 3, 4, 1, 2, 2, 1, 4, 4, 0, 0, 0, 0, 4, 5, 2, 1, 1, 2, 5, 5, 0, 2, 2, 0, 5, 6, 2, 1, 0, 0, 1, 2, 6, 6, 0, 0, 0, 0, 0, 0, 6, 7, 2, 4, 2, 2, 4, 2, 7, 7, 0, 0, 0, 0, 0, 0, 7, 8, 3, 1, 0, 0, 1, 3, 8, 8, 0, 2, 1, 2, 2, 1, 2, 0, 8, 9, 3, 2, 1, 1, 1, 1, 2, 3, 9
Offset: 1

Views

Author

Hartmut F. W. Hoft, Sep 09 2015

Keywords

Comments

This sequence lists the areas of rectangular pieces computed from the lengths of the legs in the Dyck path for n given in the n-th row of A237593 and the widths between the legs of the (n-1)-st and n-th Dyck paths given in the n-th row of A262045. The sum of the areas of all pieces appears to equal sigma(n), adjusting for a double count of the width at the center (that matters when the two symmetric Dyck paths enclose an odd number of regions).
T(n,k) is the sum of the widths of the symmetric representation of sigma(n) that are associated with the k-th line segment of the n-th Dyck path of the original diagram of the symmetric representation of sigma in the first quadrant after the diagram has been partitioned into two octants whose vertices are (for example) at (0, 1) and (1, 0). Note that the new diagram contains two main diagonals that are parallel between them, see example. - Omar E. Pol, Sep 30 2015

Examples

			For the 9th row: [5 2 2 2 2 5] in A237593 and [1 0 1 1 0 1] in A262045 result in [5 0 2 2 0 5] in this sequence which defines three regions between the Dyck paths for 8 and 9. The areas of the three regions are 5, 3 and 5; note that the width at the center of the middle region is 1 and is counted twice when adding up contiguous subsequences of nonzero numbers. The sum of the three areas is sigma(9) = 13.
For the 19th row: [10 4 2 2 1 1 2 2 4 10] in A237593 and [1 0 0 0 0 0 0 0 0 1] in A262045 result in [10 0 0 0 0 0 0 0 0 10] in this sequence which defines two disjoint regions of area 10 each between the Dyck paths for 18 and 19; their sum is sigma(19) = 20.
The first 20 rows of the triangle:
  -------------------------------------
  n\k   1  2  3  4  5  6  7  8  9  10
  -------------------------------------
  1:    1, 1;
  2:    2, 2;
  3:    2, 0, 0, 2;
  4:    3, 1, 1, 3;
  5:    3, 0, 0, 3;
  6:    4, 1, 2, 2, 1, 4;
  7:    4, 0, 0, 0, 0, 4;
  8:    5, 2, 1, 1, 2, 5;
  9:    5, 0, 2, 2, 0, 5;
  10:   6, 2, 1, 0, 0, 1, 2, 6;
  11:   6, 0, 0, 0, 0, 0, 0, 6;
  12:   7, 2, 4, 2, 2, 4, 2, 7;
  13:   7, 0, 0, 0, 0, 0, 0, 7;
  14:   8, 3, 1, 0, 0, 1, 3, 8;
  15:   8, 0, 2, 1, 2, 2, 1, 2, 0,  8;
  16:   9, 3, 2, 1, 1, 1, 1, 2, 3,  9;
  17:   9, 0, 0, 0, 0, 0, 0, 0, 0,  9;
  18:  10, 3, 4, 2, 1, 1, 2, 4, 3, 10;
  19:  10, 0, 0, 0, 0, 0, 0, 0, 0, 10;
  20:  11, 4, 2, 1, 4, 4, 1, 2, 4, 11;
  ...
A diagram of the first six rows shows the regions with an "x" marking the cells on the diagonal that are double counted.
.                                        _ _ _ _
.                                       |_|_|_|_|_
.                                             |_|x|_
.                                             |x|_|_|
.                  [4 1 2 2 1 4] & sigma(6)=12    |_|
.                                                 |_|
.                             _ _ _               |_|
.                            |_|_|_|
.                                  |_ _
.           [3 0 0 3] & sigma(5)=6   |_|
.                                    |_|
.                    _ _ _           |_|
.                   |_|_|_|_
.                       |x|_|
.                         |_| [3 1 1 3] & sigma(4)=7
.             _ _         |_|
.            |_|_|_
.                |_|    [2 0 0 2] & sigma(3)=4
.                |_|
.      _ _
.     |_ x|    [2 2] & sigma(2)=3
.       |_|
.  _
. |x|    [1 1] & sigma(1)=1
.
From _Omar E. Pol_, Sep 30 2015: (Start)
Illustration of the 12th row of the triangle:
.
.           7
.     _ _ _ _ _ _ _  2
.     / / / / / / /|   4
.                 /|_ _  2
.                 / / /|
.                 / / /      2
.                   /        _ 4
.                          / /|  2
.                        / / /|_ _
.                          / / / /|
.                                /|
.                                /|
.                                /| 7
.                                /|
.                                /|
.                                /|
.
The original diagram of the symmetric representation of sigma(12) in the first quadrant has been partitioned into two octants. Note that now there are two main diagonals in the diagram. The sums of the widths associated to the successive line segments of the partitioned zig-zag path give [7, 2, 4, 2, 2, 4, 2, 7], the same as the 12th row of the triangle.
(End)

Crossrefs

Cf. A000203, A003056, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A262045.

Programs

  • Mathematica
    (* auxiliary functions are defined in A237048, A237593 and A262045 respectively *)
s[n_,k_]:=Ceiling[(n+1)/k-(k+1)/2] - Ceiling[(n+1)/(k+1)-(k+2)/2]
t[n_,k_]:=If[k<=row[n], s[n, k], s[n, 2*row[n]+1-k]]
a262048[n_]:=Map[t[n, #]&, Range[2*row[n]]] a262045[n]
Flatten[Map[a262048, Range[16]]] (* data *)

A282197 a(n) is the smallest number d if the point (d,d) is shared by exactly n 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, 19818, 18915, 60349, 78369, 113010, 110185, 91650, 85171, 311321, 123788, 823049, 128596, 1650408, 1136865, 415355, 906771, 2897535
Offset: 1

Views

Author

Hartmut F. W. Hoft, Feb 08 2017

Keywords

Comments

This sequence is not monotone since a(19) = 19818 > 18915 = a(20).
Additional values smaller than 5000000 are a(37) = 1751785, a(38) = 1786732, a(39) = 1645139, a(41) = 1308771 and a(44) = 3329668.
Sequence A128605 of first occurrences of gaps between adjacent Dyck paths appears to be unrelated to this sequence.
First differs from A279286 (which is monotone) at a(19). - Omar E. Pol, Feb 08 2017
a(n) = d if the point (d,d) belongs to the first vertical-line-segment of exactly length n found 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 pyramid. - Omar E. Pol, Feb 09 2017

Examples

			The four examples listed in A279286 are also examples for this sequences.
a(20) = 18915 is in the sequence since it is the first time that exactly 20 Dyck paths meet on the diagonal though a concurrence of exactly 19 paths on the diagonal happens only later at a(19) = 19818.

Crossrefs

Cf. A128605, A237593, A240542, A245092, A279286.

Programs

  • Mathematica
    a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
(* parameter cL must be sufficiently large for bound b *)
a282197[cL_, b_] := Module[{centers=Map[0&, Range[cL]], 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[centers[[len]]==0, centers[[len]]=cPrev]; acc={cCur}; cPrev=cCur]; k++]; centers]
a282197[50, 5000000] (* data *)
(* list processing implementation useful for "small" arguments only *)
a282197F[n_] := Map[Last, Sort[Normal[Map[First[First[#]]&, GroupBy[Split[Map[a240542, Range[n]]], Length[#]&]]]]]
a282197F[50000] (* computes a(1) .. a(20) *)

A380580 Irregular tetrahedron T(s,r,k) read by rows in which the slice s is an irregular triangle, itself read by rows, in which row r lists the r-th row of A237593 sandwiched between two A380579(s+1,r+1), with s >= 0; 0 <= r <= s; k >= 0. Assume that row 0 of A237593 is empty.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 1, 1, 4, 4, 3, 1, 1, 3, 2, 2, 2, 2, 5, 5, 4, 1, 1, 4, 3, 2, 2, 3, 2, 2, 1, 1, 2, 2, 7, 7, 6, 1, 1, 6, 5, 2, 2, 5, 4, 2, 1, 1, 2, 4, 3, 3, 1, 1, 3, 3, 8, 8, 7, 1, 1, 7, 6, 2, 2, 6, 5, 2, 1, 1, 2, 5, 4, 3, 1, 1, 3, 4, 3, 3, 2, 2, 3, 3, 10, 10, 9, 1, 1, 9, 8, 2, 2, 8, 7, 2, 1, 1, 2, 7, 6, 3, 1, 1, 3, 6, 5, 3, 2, 2, 3, 5
Offset: 0

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Author

Omar E. Pol, Jan 27 2025

Keywords

Comments

The discussion of this sequence was too long to be included here, and can be found in the attached "Discussion" text file (see the first link). - N. J. A. Sloane, Jul 31 2025

Links

Crossrefs

See the "Discussion" text file for the cross-references.

Programs

  • Mathematica
    A237593row[n_] := Join[#, Reverse[#]] & [Table[Ceiling[(n+1)/k - (k+1)/2] + Quotient[k*(k+3) - 2*n, 2*(k+1)], {k, Quotient[Sqrt[8*n + 1] - 1, 2]}]];
A380580slice[s_] := Table[Join[#, A237593row[r], #] & [{Quotient[3*s, 2] - r + 1}], {r, 0, s}];
Array[A380580slice, 10, 0] (* Paolo Xausa, Aug 19 2025 *)

Extensions

Edited by N. J. A. Sloane, Jul 31 2025
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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