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.

Showing 1-3 of 3 results.

A302248 Irregular triangle read by rows in which the odd-indexed terms of the n-th row together with the even-indexed terms of the same row but listed in reverse order give the n-th row of triangle A299778.

Original entry on oeis.org

1, 3, 2, 2, 7, 0, 3, 3, 12, 0, 0, 4, 4, 0, 15, 0, 0, 5, 5, 3, 9, 0, 0, 9, 6, 6, 0, 0, 28, 0, 0, 0, 7, 7, 0, 0, 12, 0, 0, 12, 8, 8, 8, 0, 0, 31, 0, 0, 0, 0, 9, 9, 0, 0, 0, 39, 0, 0, 0, 0, 10, 10, 0, 0, 0, 42, 0, 0, 0, 0, 11, 11, 5, 0, 0, 5, 18, 0, 0, 18, 0, 0, 12, 12, 0, 0, 0, 0, 60, 0, 0, 0, 0, 0, 13, 13, 0, 0, 5, 0
Offset: 1

Views

Author

Omar E. Pol, Apr 10 2018

Keywords

Examples

			Triangle begins (rows 1..28):
   1;
   3;
   2,  2;
   7,  0;
   3,  3;
  12,  0,  0;
   4,  4,  0;
  15,  0,  0;
   5,  5,  3;
   9,  0,  0,  9;
   6,  6,  0,  0;
  28,  0,  0,  0;
   7,  7,  0,  0;
  12,  0,  0, 12;
   8,  8,  8,  0,  0;
  31,  0,  0,  0,  0;
   9,  9,  0,  0,  0;
  39,  0,  0,  0,  0;
  10, 10,  0,  0,  0;
  42,  0,  0,  0,  0;
  11, 11,  5,  0,  0,  5;
  18,  0,  0, 18,  0,  0;
  12, 12,  0,  0,  0,  0;
  60,  0,  0,  0,  0,  0;
  13, 13,  0,  0,  5,  0;
  21,  0,  0, 21,  0,  0;
  14, 14,  6,  0,  0,  6;
  56,  0,  0,  0,  0,  0,  0;
...
For n = 21 the 21st row of A299778 is [11, 5, 0, 5, 0, 11], so the 21st row of this triangle is [11, 11, 5, 0, 0, 5].

Links

Crossrefs

Row sums give A000203.
Row n has length A003056(n).
Column k starts in row A000217(k).
The number of nonzero terms in row n is A237271(n).
Column 1 is A241838.
The triangle with n rows contain A237590(n) nonzero terms.
Cf. A280850 (analog for subparts).
Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239660, A240542, A244050, A245092, A262626, A279387, A279388, A280851, A299778.

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.

A356351 Partial sums of the ziggurat sequence A347186.

Original entry on oeis.org

1, 5, 11, 27, 39, 76, 96, 160, 196, 286, 328, 489, 545, 701, 808, 1064, 1154, 1488, 1598, 2006, 2208, 2550, 2706, 3403, 3610, 4072, 4384, 5169, 5409, 6385, 6657, 7681, 8127, 8883, 9324, 10910, 11290, 12220, 12824, 14560, 15022, 16863, 17369, 19175, 20276, 21608, 22208, 25129, 25849, 27669
Offset: 1

Views

Author

Omar E. Pol, Oct 15 2022

Keywords

Comments

a(n) is the volume (or the number of cubes) in a polycube whose base is the symmetric representation of A024916(n) which is formed with the first n 3D-Ziggurats described in A347186.
a(n) is also the total number of cubes in a three-dimensional spiral formed with the first n 3D-Ziggurats described in A347186 (see example). The base of the 3D-spiral is the spiral formed with the symmetric representation of sigma of the first n positive integers as shown in the example section of A239660.

Examples

			For n = 16 the figure shows the top view of a three-dimensional spiral formed with the first 16 3D-Ziggurats described in A347186. There are four 3D-Ziggurats in every quadrant:
.
                  _ _ _ _ _ _ _ _
                 |_|_|_|_|_|_|_|_|_ _ _ _ _ _ _
                 |_|             |_|_|_|_|_|_|_|
                _|_|                           |
               |_|_|  _ _ _ _ _ _              |_ _
            _ _|     |_|_|_|_|_|_|_ _ _ _ _        |_
      _ _ _|_|      _|_|         |_|_|_|_|_|         |
     |_|_|_|_|    _|_|_|                   |_ _      |_ _ _
     |_|      _ _|_|      _ _ _ _          |_|_|         |_|
     |_|     |_|_|_|    _|_|_|_|_|_ _ _      |_|_ _      |_|
     |_|     |_|      _|_|_|     |_|_|_|         |_|     |_|
     |_|     |_|     |_|_|_|           |_ _      |_|     |_|
     |_|     |_|     |_|      _ _        |_|     |_|     |_|
     |_|     |_|     |_|     |_|_|_      |_|     |_|     |_|
    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
   |_|     |_|     |_|     |_|         |_|     |_|     |_|     |_|
   |_|     |_|     |_|     |_|_ _     _|_|     |_|     |_|     |_|
   |_|     |_|     |_|       |_|_|_ _|_|_|     |_|     |_|     |_|
   |_|     |_|     |_|_          |_|_|_|    _ _|_|     |_|     |_|
   |_|     |_|         |_                 _|_|_|_|     |_|     |_|
   |_|     |_|_ _        |_ _ _ _        |_|_|    _ _ _|_|     |_|
   |_|           |_      |_|_|_|_|_ _ _ _|_|    _|_|_|_|_|     |_|
   |_|_ _ _        |_            |_|_|_|_|_|  _|_|_|_|    _ _ _|_|
         |_|_ _      |                       |_|_|_|_|   |_|_|_|_|
         |_|_|_|     |_ _ _ _ _ _            |_|_|_|    _|_|
           |_|_|_    |_|_|_|_|_|_|_ _ _ _ _ _|_|      _|_|_|
             |_|_|               |_|_|_|_|_|_|_|  _ _|_|_|
                 |                               |_|_|_|
                 |_ _ _ _ _ _ _ _                |_|
                 |_|_|_|_|_|_|_|_|_ _ _ _ _ _ _ _|_|
                                 |_|_|_|_|_|_|_|_|_|
.
The number of square cells in the top view of the n-th 3D-Ziggurat equals A000203(n).
The total number of square cells in the top view of the 3D-Spiral with the first n 3D-Ziggurats equals A024916(n).
In the above figure the total number of square cells equals A024916(16) = 220.
a(16) = 1064 is the total number of cubes in the 3D-Spiral with the first 16 3D-Ziggurats.

Crossrefs

Cf. A000203, A024916, A196020, A235791, A236104, A237270, A237591, A237593, A239660, A239931, A239932, A239933, A239934, A347186, A296508, A299778, A347186, A347263, A347367, A347529, A351819.
Showing 1-3 of 3 results.

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