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A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 18, 19, 21, 23, 26, 27, 29, 30, 32, 33, 37, 39, 41, 42, 45, 47, 51, 52, 54, 55, 57, 58, 62, 64, 67, 68, 70, 72, 76, 77, 79, 80, 82, 84, 87, 89, 91, 92, 95, 98, 102, 104, 106, 107, 111, 112, 116, 118, 120, 121, 123, 125, 130, 131, 135, 136, 138, 140, 144, 147, 149, 150, 152, 154
Offset: 1

Views

Author

Omar E. Pol, Mar 31 2014

Keywords

Comments

The total area (or total number of cells) of the diagram after n stages is equal to A024916(n), the sum of all divisors of all positive integers <= n.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(n), see example.
For more information see A237593 and A237270.
a(n) is also the total number of terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Apr 20 2016

Examples

			Illustration of initial terms:
.                                                         _ _ _ _
.                                           _ _ _        |_ _ _  |_
.                               _ _ _      |_ _ _|       |_ _ _|   |_
.                     _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.             _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.       _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.      |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.
.       1      2        4          5            7              8
.
For n = 6 the diagram contains 8 regions (or parts), so a(6) = 8.
The sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 33. On the other hand after 6 stages the sum of all parts of the diagram is [1] + [3] + [2+2] + [7] + [3+3] + [12] = 33, equaling the sum of all divisors of all positive integers <= 6.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(6) = 3.
From _Omar E. Pol_, Dec 25 2020: (Start)
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
(End)

Links

Crossrefs

Partial sums of A237271.
Compare with A060831 (analog for the diagram that contains subparts).
Cf. A000203, A004125, A024916, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239659, A239660, A239663, A239665, A239931-A239934, A245092, A244050, A244970, A262626, A317109.

Programs

  • Mathematica
    (* total number of parts in the first n symmetric representations *)
(* Function a237270[] is defined in A237270 *)
(* variable "previous" represents the sum from 1 through m-1 *)
a237590[previous_,{m_,n_}]:=Rest[FoldList[Plus[#1,Length[a237270[#2]]]&,previous,Range[m,n]]]
a237590[n_]:=a237590[0,{1,n}]
a237590[78] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)

Formula

a(n) = A317109(n) - A294723(n) + 1 (Euler's formula). - Omar E. Pol, Jul 21 2018

Extensions

Definition clarified by Omar E. Pol, Jul 21 2018

A262612 Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A236104.

Original entry on oeis.org

1, 5, 14, 1, 30, 2, 55, 6, 91, 10, 1, 140, 19, 2, 204, 28, 3, 285, 44, 7, 385, 60, 11, 1, 506, 85, 15, 2, 650, 110, 24, 3, 819, 146, 33, 4, 1015, 182, 42, 8, 1240, 231, 58, 12, 1, 1496, 280, 74, 16, 2, 1785, 344, 90, 20, 3, 2109, 408, 115, 29, 4, 2470, 489, 140, 38, 5, 2870, 570, 165, 47, 9, 3311, 670, 201, 56, 13, 1
Offset: 1

Views

Author

Omar E. Pol, Nov 03 2015

Keywords

Comments

Alternating sum of row n equals A175254(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A175254(n), which is also the volume (or the total number of units cubes) in the first n levels of the stepped pyramid described in A245092.
Row n has length A003056(n) hence the first element of column k is in row A000217(k).

Examples

			Triangle begins:
     1;
     5;
    14,    1;
    30,    2;
    55,    6;
    91,   10,    1;
   140,   19,    2;
   204,   28,    3;
   285,   44,    7;
   385,   60,   11,    1;
   506,   85,   15,    2;
   650,  110,   24,    3;
   819,  146,   33,    4;
  1015,  182,   42,    8;
  1240,  231,   58,   12,    1;
  1496,  280,   74,   16,    2;
  1785,  344,   90,   20,    3;
  2109,  408,  115,   29,    4;
  2470,  489,  140,   38,    5;
  2870,  570,  165,   47,    9;
  3311,  670,  201,   56,   13,    1;
  3795,  770,  237,   72,   17,    2;
  4324,  891,  273,   88,   21,    3;
  4900, 1012,  322,  104,   25,    4;
  ...
For n = 6 we have that A175254(6) = [1] + [1 + 3] + [1 + 3 + 4] + [1 + 3 + 4 + 7] + [1 + 3 + 4 + 7 + 6] + [1 + 3 + 4 + 7 + 6 + 12] = 1 + 4 + 8 + 15 + 21 + 33 = 82. On the other hand the alternating sum of the 6th row of the triangle is 91 - 10 + 1 = 82, equaling A175254(6).

Crossrefs

Cf. A000203, A000217, A003056, A024916, A175254, A196020, A235791, A236104, A237048, A237591, A237593, A237270, A237271, A245092, A261699, A262626.
Column 1 gives A000330, n >= 1. Column 2 is A005993. It appears that column 3 is A092353.

A294723 a(n) is the total number of vertices after n-th stage in the diagram of the symmetries of sigma described in A236104, with a(0) = 1.

Original entry on oeis.org

1, 4, 7, 11, 16, 20, 27, 31, 38, 45, 53, 57, 66, 70, 78, 89, 100, 104, 115, 119, 130, 142, 150, 154, 167, 176, 184, 196, 211, 215, 230, 234, 249, 261, 269, 280, 297, 301, 309, 321, 338, 342, 359, 363, 379, 398, 406, 410, 429, 440, 459, 471, 487, 491, 510
Offset: 0

Views

Author

Omar E. Pol, Nov 07 2017

Keywords

Comments

a(n) is also the total number of "hinges" in the "mechanism" where every row of the two-dimensional diagram of the isosceles triangle with n rows described in A237593 is folded in a 90-degree zig-zag, appearing the structure of the stepped pyramid with n levels described in A245092. Note that the diagram described in A236104 is also the top view of the mentioned pyramid. The area of the terraces in the n-th level of the pyramid, starting from the top, equals sigma(n) = A000203(n).
For the construction of the two-dimensional diagram using Dyck paths and for more information about the pyramid see A237593 and A262626.
Note that every line segment of the Dyck paths of the diagram is related to partitions into consecutive parts (see A237591). - Omar E. Pol, Feb 23 2018

Examples

			Illustration of initial terms (n = 0..9):
.                                                           _ _ _ _
.                                             _ _ _        |_ _ _  |_
.                                 _ _ _      |_ _ _|       |_ _ _|   |_
.                       _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.               _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.         _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    .   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.    1    4      7        11         16           20             27
.
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _|   |_ _      |_ _ _|   |_ _  |     |_ _ _|   |_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           31                  38                     45
.
.
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.

Links

Crossrefs

Cf. A317109 (number of edges).
Cf. A237590 (number of regions or parts).
Compare with A317293 (analog for the diagram that contains subparts).
Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A294847, A296508.

Formula

a(n) = A317109(n) - A237590(n) + 1 (Euler's formula). - Omar E. Pol, Jul 21 2018

Extensions

Terms a(30) and beyond from Robert Price, Jul 31 2018
Example extended for a(7)-a(9) and a(29) by Omar E. Pol, Jul 31 2018

A317109 a(n) is the total number of edges after n-th stage in the diagram of the symmetries of sigma described in A236104, with a(0) = 0.

Original entry on oeis.org

0, 4, 8, 14, 20, 26, 34, 40, 48, 58, 68, 74, 84, 90, 100, 114, 126, 132, 144, 150, 162, 178, 188, 194, 208, 220, 230, 246, 262, 268, 284, 290, 306, 322, 332, 346, 364, 370, 380, 396, 414, 420, 438, 444, 462, 484, 494, 500, 520, 534, 556, 572, 590, 596, 616, 636
Offset: 0

Views

Author

Omar E. Pol, Jul 21 2018

Keywords

Comments

All terms are even numbers.
Note that the two-dimensional diagram is also the top view of the stepped pyramid with n levels described in A245092.
For the construction of the two-dimensional diagram using Dyck paths and for more information about the pyramid see A237593.

Examples

			Illustration of initial terms (n = 1..9):
.                                                       _ _ _ _
.                                         _ _ _        |_ _ _  |_
.                             _ _ _      |_ _ _|       |_ _ _|   |_
.                   _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.           _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.     _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.     4      8        14         20           26             34
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _|   |_ _      |_ _ _|   |_ _  |     |_ _ _|   |_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           40                  48                     58
.
.
Illustration of the diagram after 29 stages (contain 268 edges, 215 vertices and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.

Links

Crossrefs

Cf. A294723 (number of vertices).
Cf. A237590 (number of regions or parts).
Compare with A317292 (analog for the diagram that contains subparts).
Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A296508.

Formula

a(n) = A294723(n) + A237590(n) - 1 (Euler's formula).

Extensions

More terms and b-file from Robert Price, Jul 31 2018

A294847 a(n) is the total number of line segments in the diagram described in A236104 after n-th stage, with a(0) = 2.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 20, 22, 28, 32, 38, 40, 48, 50, 56, 64, 74, 76, 86, 88, 98, 106, 112, 114, 126, 132, 138, 146, 160, 162
Offset: 0

Views

Author

Omar E. Pol, Nov 09 2017

Keywords

Comments

Note that the diagram is also the top view of the stepped pyramid with n levels described in A245092.
For the construction of the diagram using Dyck paths and for more information about the pyramid see A237593.

Examples

			Illustration of initial terms (n = 1..6):
.                                                      _ _ _ _
.                                        _ _ _        |_ _ _  |_
.                            _ _ _      |_ _ _|       |_ _ _|   |_
.                  _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.          _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.    _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.    4      6        8          12           14             20

Crossrefs

Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237590, A237591, A237593, A245092, A262626, A294723.

A293750 a(n) is the total number of line segments that belong to the Dyck paths in the diagram of the symmetries of sigma described in A236104 and A237593 after n-th stage, with a(0) = 0.

Original entry on oeis.org

0, 2, 4, 6, 10, 12, 18, 20, 26, 30, 36, 38, 46, 48, 54, 62, 72, 74, 84, 86, 96, 104, 110, 112, 124, 130, 136, 144, 158, 160
Offset: 0

Views

Author

Omar E. Pol, Nov 09 2017

Keywords

Examples

			Illustration of initial terms (n = 1..6):
.                                                      _ _ _ _
.                                        _ _ _         _ _ _  |_
.                            _ _ _       _ _ _|        _ _ _|   |_
.                  _ _       _ _  |_     _ _  |_ _     _ _  |_ _  |
.          _ _     _ _|_     _ _|_  |    _ _|_  | |    _ _|_  | | |
.    _     _  |    _  | |    _  | | |    _  | | | |    _  | | | | |
.     |     | |     | | |     | | | |     | | | | |     | | | | | |
.
.    2      4        6          10           12             18
.

Crossrefs

Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237590, A237591, A237593, A245092, A262626, A294723, A294847.

Formula

a(n) = A294847(n) - 2.

A294848 a(n) is the total number of horizontal line segments in the diagram of the symmetries of sigma described in A236104 after n-th stage, with a(0) = 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 11, 14, 16, 19, 20, 24, 25, 28, 32, 37, 38, 43, 44, 49, 53, 56, 57, 63, 66, 69, 73, 80, 81
Offset: 0

Views

Author

Omar E. Pol, Nov 09 2017

Keywords

Comments

Since the diagram is symmetrics a(n) is also the number of vertical line segments after n-th stage. For a diagram see A294847.

Crossrefs

Cf. A196020, A235791, A236104, A236106, A237591, A237593, A245092, A293750, A294847, A294849.

Formula

a(n) = A294847(n)/2.

A294849 a(n) is the total number of horizontal line segments that belong to the Dyck paths in the diagram of the symmetries of sigma described in A236104 and A237593 after n-th stage, with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 10, 13, 15, 18, 19, 23, 24, 27, 31, 36, 37, 42, 43, 48, 52, 55, 56, 62, 65, 68, 72, 79, 80
Offset: 0

Views

Author

Omar E. Pol, Nov 09 2017

Keywords

Comments

Since the diagram is symmetrics a(n) is also the number of vertical line segments after n-th stage. For a diagram see A293750.

Crossrefs

Cf. A000203, A196020, A235791, A236104, A237048, A237591, A237593, A245092, A293750, A294847, A294848.

Formula

a(n) = A293750(n)/2.

A339576 Row sums of triangle A236104.

Original entry on oeis.org

1, 4, 10, 17, 29, 41, 59, 74, 101, 121, 151, 179, 215, 245, 295, 326, 374, 423, 477, 519, 591, 641, 707, 767, 844, 904, 1000, 1056, 1140, 1234, 1324, 1387, 1507, 1587, 1701, 1794, 1902, 1992, 2136, 2226, 2346, 2476, 2602, 2692, 2874, 2984, 3122, 3246, 3397
Offset: 1

Views

Author

N. J. A. Sloane, Dec 18 2020

Keywords

Comments

In other words, a(n) is the sum of squares of terms in row n of A235791.

Crossrefs

Cf. A024916, A060831, A235791, A236104, A237593, A240542.

Programs

  • Maple
    a:= n-> add(ceil((n+1)/k-(k+1)/2)^2, k=1..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 18 2020
  • PARI
    a(n) = sum(k=1, floor((sqrt(8*n+1)-1)/2), ceil((n+1)/k-(k+1)/2)^2) \\ Felix Fröhlich, Dec 19 2020; adapted from Maple code

A237593 Triangle read by rows in which row n lists the elements of the n-th row of A237591 followed by the same elements in reverse order.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Feb 22 2014

Keywords

Comments

Row n is a palindromic composition of 2*n.
T(n,k) is also the length of the k-th segment in a Dyck path on the first quadrant of the square grid, connecting the x-axis with the y-axis, from (n, 0) to (0, n), starting with a segment in vertical direction, see example.
Conjecture 1: the area under the n-th Dyck path equals A024916(n), the sum of all divisors of all positive integers <= n.
If the conjecture is true then the n-th Dyck path represents the boundary segments after the alternating sum of the elements of the n-th row of A236104.
Conjecture 2: two adjacent Dyck paths never cross (checked by hand up to n = 128), hence the total area between the n-th Dyck path and the (n-1)-st Dyck path is equal to sigma(n) = A000203(n), the sum of divisors of n.
The connection between A196020 and A237271 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> this sequence --> A239660 --> A237270 --> A237271.
PARI scripts area(n) and chkcross(n) have been written to check the 2 properties and have been run up to n=10000. - Michel Marcus, Mar 27 2014
Mathematica functions have been written that verified the 2 properties through n=30000. - Hartmut F. W. Hoft, Apr 07 2014
Comments from Franklin T. Adams-Watters on sequences related to the "symmetric representation of sigma" in A235791 and related sequences, Mar 31 2014: (Start)
The place to start is with A235791, which is very simple. Then go to A237591, also very simple, and A237593, still very simple.
You then need to interpret the rows of A237593 as Dyck paths. This interpretation is in terms of run lengths, so 2,1,1,2 means up twice, down once, up once, and down twice. Because the rows of A237593 are symmetric and of even length, this path will always be symmetric.
Now the surprising fact is that the areas enclosed by the Dyck path for n (laid on its side) always includes the area enclosed for n-1; and the number of squares added is sigma(n).
Finally, look at the connected areas enclosed by n but not by n-1; the size of these areas is the symmetric representation of sigma. (End)
The symmetric representation of sigma, so defined, is row n of A237270. - Peter Munn, Jan 06 2025
It appears that, for the n-th set, the number of cells lying on the first diagonal is equal to A067742(n), the number of middle divisors of n. - Michel Marcus, Jun 21 2014
Checked Michel Marcus's conjecture with two Mathematica functions up to n=100000, for more information see A240542. - Hartmut F. W. Hoft, Jul 17 2014
A003056(n) is also the number of peaks of the Dyck path related to the n-th row of triangle. - Omar E. Pol, Nov 03 2015
The number of peaks of the Dyck path associated to the row A000396(n) of this triangle equals the n-th Mersenne prime A000668(n), hence Mersenne primes are visible in two ways at the pyramid described in A245092. - Omar E. Pol, Dec 19 2016
The limit as n approaches infinity (area under the Dyck path described in the n-th row of triangle divided by n^2) equals Pi^2/12 = zeta(2)/2. (Cf. A072691.) - Omar E. Pol, Dec 18 2021
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

Examples

			Triangle begins:
   n
   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;
  17 |  9, 4, 2, 1, 1, 1, 1, 2, 4, 9;
  18 | 10, 3, 2, 2, 1, 1, 2, 2, 3, 10;
  19 | 10, 4, 2, 2, 1, 1, 2, 2, 4, 10;
  20 | 11, 4, 2, 1, 2, 2, 1, 2, 4, 11;
  21 | 11, 4, 3, 1, 1, 1, 1, 1, 1, 3, 4, 11;
  22 | 12, 4, 2, 2, 1, 1, 1, 1, 2, 2, 4, 12;
  23 | 12, 5, 2, 2, 1, 1, 1, 1, 2, 2, 5, 12;
  24 | 13, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 13;
  ...
Illustration of rows 8 and 9 interpreted as Dyck paths in the first quadrant and the illustration of the symmetric representation of sigma(9) = 5 + 3 + 5 = 13, see below:
.
y                       y
.                       .
.                       ._ _ _ _ _                _ _ _ _ _ 5
._ _ _ _ _              .         |              |_ _ _ _ _|
.         |             .         |_ _                     |_ _ 3
.         |_            .             |                    |_  |
.           |_ _        .             |_ _                   |_|_ _ 5
.               |       .                 |                      | |
.   Area = 56   |       .    Area = 69    |          Area = 13   | |
.               |       .                 |                      | |
.               |       .                 |                      | |
. . . . . . . . | . x   . . . . . . . . . | . x                  |_|
.
.    Fig. 1                    Fig. 2                  Fig. 3
.
Figure 1. For n = 8 the 8th row of triangle is [5, 2, 1, 1, 2, 5] and the area under the symmetric Dyck path is equal to A024916(8) = 56.
Figure 2. For n = 9 the 9th row of triangle is [5, 2, 2, 2, 2, 5] and the area under the symmetric Dyck path is equal to A024916(9) = 69.
Figure 3. The symmetric representation of sigma(9): between both symmetric Dyck paths there are three regions (or parts) of sizes [5, 3, 5].
The sum of divisors of 9 is 1 + 3 + 9 = A000203(9) = 13. On the other hand the difference between the areas under the Dyck paths equals the sum of the parts of the symmetric representation of sigma(9) = 69 - 56 = 5 + 3 + 5 = 13, equaling the sum of divisors of 9.
.
Illustration of initial terms as Dyck paths in the first quadrant:
(row n = 1..28)
.  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
  |_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
  |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  |_ _ _ _ _ _ _ _ _ _ _ _ _  | |
  |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
  |_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
  |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
  |_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
  |_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
  |_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
  |_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
  |_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
  |_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
  |_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _
  |_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  |
  |_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | |
  |_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | |
  |_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | |
  |_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | |
  |_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | |
  |_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | |
  |_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | |
  |_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | |
  |_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | |
  |_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | |
  |_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | |
  |_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | |
  |_  | | | | | | | | | | | | | | | | | | | | | | | | | | |
  |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
n: 1 2 3 4 5 6 7 8 9 10..12..14..16..18..20..22..24..26..28
.
It appears that the total area (also the total number of cells) in the first n set of symmetric regions of the diagram is equal to A024916(n), the sum of all divisors of all positive integers <= n.
It appears that the total area (also the total number of cells) in the n-th set of symmetric regions of the diagram is equal to sigma(n) = A000203(n) (checked by hand up n = 128).
From _Omar E. Pol_, Aug 18 2015: (Start)
The above diagram is also the top view of the stepped pyramid described in A245092 and it is also the top view of the staircase described in A244580, in both cases the figure represents the first 28 levels of the structure. Note that the diagram contains (and arises from) a hidden pattern which is shown below.
.
Illustration of initial terms as an isosceles triangle:
Row                                 _ _
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|
...
This diagram is the simpler representation of the sequence.
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).
Note that this symmetric pattern also emerges from the front view of the stepped pyramid described in A245092, which is related to sigma A000203, the sum-of-divisors function, and other related sequences. The diagram represents the first 16 levels of the pyramid. (End)

Links

Crossrefs

Row n has length 2*A003056(n).
Row sums give A005843, n >= 1.
Column k starts in row A008805(k-1).
Column 1 = right border = A008619, n >= 1.
Bisections are in A259176, A259177.
For further information see A262626.
Cf. A000203, A000217, A001065, A001227, A024916, A048050, A054844, A067742, A072691, A131507, A196020, A221529, A224613, A235791, A236104, A237048, A237270, A237271, A237590, A237591, A239660, A239931-A239934, A244050, A244580, A245092, A249351, A261350, A261699, A262611, A262612, A279387, A280850, A280851, A286000, A286001, A296508, A335616, A340035.

Programs

  • Mathematica
    row[n_]:=Floor[(Sqrt[8n+1]-1)/2]
s[n_,k_]:=Ceiling[(n+1)/k-(k+1)/2]-Ceiling[(n+1)/(k+1)-(k+2)/2]
f[n_,k_]:=If[k<=row[n],s[n,k],s[n,2 row[n]+1-k]]
TableForm[Table[f[n,k],{n,1,50},{k,1,2 row[n]}]] (* Hartmut F. W. Hoft, Apr 08 2014 *)
  • PARI
    row(n) = {my(orow = row237591(n)); vector(2*#orow, i, if (i <= #orow, orow[i], orow[2*#orow-i+1]));}
area(n) = {my(rown = row(n)); surf = 0; h = n; odd = 1; for (i=1, #row, if (odd, surf += h*rown[i], h -= rown[i];); odd = !odd;); surf;}
heights(v, n) = {vh = vector(n); ivh = 1; h = n; odd = 1; for (i=1, #v, if (odd, for (j=1, v[i], vh[ivh] = h; ivh++), h -= v[i];); odd = !odd;); vh;}
isabove(hb, ha) = {for (i=1, #hb, if (hb[i] < ha[i], return (0));); return (1);}
chkcross(nn) = {hga = concat(heights(row(1), 1), 0); for (n=2, nn, hgb = heights(row(n), n); if (! isabove(hgb, hga), print("pb cross at n=", n)); hga = concat(hgb, 0););} \\ Michel Marcus, Mar 27 2014
  • 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)
for n in range(1, 11): print([T(n, k) for k in range(1, 2*row(n) + 1)]) # Indranil Ghosh, Apr 21 2017

Formula

Let j(n)= floor((sqrt(8n+1)-1)/2) then T(n,k) = A237591(n,k), if k <= j(n); otherwise T(n,k) = A237591(n,2*j(n)+1-k). - Hartmut F. W. Hoft, Apr 07 2014 (corrected by Omar E. Pol, May 31 2015)

Extensions

A minor edit to the definition. - N. J. A. Sloane, Jul 31 2025
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