A239053 -id:A239053 - cp's OEIS frontend

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

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

Omar E. Pol, Mar 24 2014

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

			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

Robert Price, Table of n, a(n) for n = 1..30016 (rows n = 1..412, flattened)

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.

A239050 a(n) = 4*sigma(n).

Original entry on oeis.org

4, 12, 16, 28, 24, 48, 32, 60, 52, 72, 48, 112, 56, 96, 96, 124, 72, 156, 80, 168, 128, 144, 96, 240, 124, 168, 160, 224, 120, 288, 128, 252, 192, 216, 192, 364, 152, 240, 224, 360, 168, 384, 176, 336, 312, 288, 192, 496, 228, 372, 288, 392, 216, 480, 288, 480, 320, 360, 240, 672, 248, 384, 416, 508

Offset: 1

Omar E. Pol, Mar 09 2014

4 times the sum of divisors of n.

a(n) is also the total number of horizontal cells in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every three-dimensional quadrant arises after the 90-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a square formed by four cells (see links and examples). - Omar E. Pol, Jul 04 2016

			For n = 4 the sum of divisors of 4 is 1 + 2 + 4 = 7, so a(4) = 4*7 = 28.
For n = 5 the sum of divisors of 5 is 1 + 5 = 6, so a(5) = 4*6 = 24.
.
Illustration of initial terms:                                    _ _ _ _ _ _
.                                           _ _ _ _ _ _          |_|_|_|_|_|_|
.                           _ _ _ _       _|_|_|_|_|_|_|_     _ _|           |_ _
.             _ _ _ _     _|_|_|_|_|_    |_|_|       |_|_|   |_|               |_|
.     _ _    |_|_|_|_|   |_|       |_|   |_|           |_|   |_|               |_|
.    |_|_|   |_|   |_|   |_|       |_|   |_|           |_|   |_|               |_|
.    |_|_|   |_|_ _|_|   |_|       |_|   |_|           |_|   |_|               |_|
.            |_|_|_|_|   |_|_ _ _ _|_|   |_|_         _|_|   |_|               |_|
.                          |_|_|_|_|     |_|_|_ _ _ _|_|_|   |_|_             _|_|
.                                          |_|_|_|_|_|_|         |_ _ _ _ _ _|
.                                                                |_|_|_|_|_|_|
.
n:     1          2             3                4                     5
S(n):  1          3             4                7                     6
a(n):  4         12            16               28                    24
.
For n = 1..5, the figure n represents the reflection in the four quadrants of the symmetric representation of S(n) = sigma(n) = A000203(n). For more information see A237270 and A237593.
The diagram also represents the top view of the first four terraces of the stepped pyramid described in Comments section. - _Omar E. Pol_, Jul 04 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Folding the first eight rows of triangle
Index entries for sequences related to sigma(n)

Alternating row sums of A239662.
Partial sums give A243980.
k times sigma(n), k=1..6: A000203, A074400, A272027, this sequence, A274535, A274536.
k times sigma(n), k = 1..10: A000203, A074400, A272027, this sequence, A274535, A274536, A319527, A319528, A325299, A326122.
Cf. A008438, A017113, A062731, A112610, A144613, A193553, A196020, A235791, A236104, A237270, A237593, A239052, A239053, A239660, A239662, A244050, A262626.

[4*SumOfDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jul 30 2019

with(numtheory): seq(4*sigma(n), n=1..64); # Omar E. Pol, Jul 04 2016

Array[4 DivisorSigma[1, #] &, 64] (* Michael De Vlieger, Nov 16 2017 *)

a(n) = 4 * sigma(n); \\ Omar E. Pol, Jul 04 2016

a(n) = 4*A000203(n) = 2*A074400(n).
a(n) = A000203(n) + A272027(n). - Omar E. Pol, Jul 04 2016
Dirichlet g.f.: 4*zeta(s-1)*zeta(s). - Ilya Gutkovskiy, Jul 04 2016
Conjecture: a(n) = sigma(3*n) = A144613(n) iff n is not a multiple of 3. - Omar E. Pol, Oct 02 2018
The conjecture above is correct. Write n = 3^e*m, gcd(3, m) = 1, then sigma(3*n) = sigma(3^(e+1))*sigma(m) = ((3^(e+2) - 1)/2)*sigma(m) = ((3^(e+2) - 1)/(3^(e+1) - 1))*sigma(3^e*m), and (3^(e+2) - 1)/(3^(e+1) - 1) = 4 if and only if e = 0. - Jianing Song, Feb 03 2019

A239933 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-1).

Original entry on oeis.org

2, 2, 4, 4, 6, 6, 8, 8, 8, 10, 10, 12, 12, 14, 6, 6, 14, 16, 16, 18, 12, 18, 20, 8, 8, 20, 22, 22, 24, 24, 26, 10, 10, 26, 28, 8, 8, 28, 30, 30, 32, 12, 16, 12, 32, 34, 34, 36, 36, 38, 24, 24, 38, 40, 40, 42, 42, 44, 16, 16, 44, 46, 20, 46, 48, 12, 12, 48, 50, 18, 20, 18, 50, 52, 52, 54, 54, 56, 20, 20, 56, 58, 14, 14, 58, 60, 12, 12, 60, 62, 22, 22, 62, 64, 64

Offset: 1

Omar E. Pol, Mar 29 2014

Row n is a palindromic composition of sigma(4n-1).
Row n is also the row 4n-1 of A237270.
Row n has length A237271(4n-1).
Row sums give A239053.
Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the third quadrant of the spiral described in A239660, see example.
For the parts of the symmetric representation of sigma(4n-3), see A239931.
For the parts of the symmetric representation of sigma(4n-2), see A239932.
For the parts of the symmetric representation of sigma(4n), see A239934.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

			The irregular triangle begins:
2, 2;
4, 4;
6, 6;
8, 8, 8;
10, 10;
12, 12;
14, 6, 6, 14;
16, 16;
18, 12, 18;
20, 8, 8, 20;
22, 22;
24, 24;
26, 10, 10, 26;
28, 8, 8, 28;
30, 30;
32, 12, 16, 12, 32;
...
Illustration of initial terms in the third quadrant of the spiral described in A239660:
.     _       _       _       _       _       _       _       _
.    | |     | |     | |     | |     | |     | |     | |     | |
.    | |     | |     | |     | |     | |     | |     | |     |_|_ _
.    | |     | |     | |     | |     | |     | |     | |    2  |_ _|
.    | |     | |     | |     | |     | |     | |     |_|_     2
.    | |     | |     | |     | |     | |     | |    4    |_
.    | |     | |     | |     | |     | |     |_|_ _        |_ _ _ _
.    | |     | |     | |     | |     | |    6      |_      |_ _ _ _|
.    | |     | |     | |     | |     |_|_ _ _        |_   4
.    | |     | |     | |     | |    8      | |_ _      |
.    | |     | |     | |     |_|_ _ _      |_    |     |_ _ _ _ _ _
.    | |     | |     | |   10        |       |_  |_    |_ _ _ _ _ _|
.    | |     | |     |_|_ _ _ _      |_ _   8  |_ _|  6
.    | |     | |   12          |         |_        |
.    | |     |_|_ _ _ _ _      |_ _        |       |_ _ _ _ _ _ _ _
.    | |   14          | |         |_      |_ _    |_ _ _ _ _ _ _ _|
.    |_|_ _ _ _ _      | |_ _        |_        |  8
.  16            |     |_ _  |         |       |
.                |         |_|_        |_ _    |_ _ _ _ _ _ _ _ _ _
.                |_ _     6    |_ _        |   |_ _ _ _ _ _ _ _ _ _|
.                    |         |_  |       | 10
.                    |_       6  | |_ _    |
.                      |_        |_ _ _|   |_ _ _ _ _ _ _ _ _ _ _ _
.                        |_ _          |   |_ _ _ _ _ _ _ _ _ _ _ _|
.                            |         | 12
.                            |_ _ _    |
.                                  |   |_ _ _ _ _ _ _ _ _ _ _ _ _ _
.                                  |   |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                  | 14
.                                  |
.                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                16
.
For n = 7 we have that 4*7-1 = 27 and the 27th row of A237593 is [14, 5, 3, 2, 1, 2, 2, 1, 2, 3, 5, 14] and the 26th row of A237593 is [14, 5, 2, 2, 2, 1, 1, 2, 2, 2, 5, 14] therefore between both Dyck paths there are four regions (or parts) of sizes [14, 6, 6, 14], so row 7 is [14, 6, 6, 14].
The sum of divisors of 27 is 1 + 3 + 9 + 27 = A000203(27) = 40. On the other hand the sum of the parts of the symmetric representation of sigma(27) is 14 + 6 + 6 + 14 = 40, equaling the sum of divisors of 27.

Cf. A000203, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239053, A239660, A239931, A239932, A239934, A244050, A245092, A262626.

A112610 Number of representations of n as a sum of two squares and two triangular numbers.

Original entry on oeis.org

1, 6, 13, 14, 18, 32, 31, 30, 48, 38, 42, 78, 57, 54, 80, 62, 84, 96, 74, 96, 121, 108, 90, 128, 98, 102, 192, 110, 114, 182, 133, 156, 176, 160, 138, 192, 180, 150, 234, 158, 192, 288, 183, 174, 240, 182, 228, 320, 194, 198, 272, 252, 240, 288, 256, 252, 403, 230

Offset: 0

James Sellers, Dec 21 2005

nonn

Also row sums of A239931, hence the sequence has a symmetric representation. - Omar E. Pol, Aug 30 2015

			a(1) = 6 since we can write 1 = 1^2 + 0^2 + 0 + 0 = (-1)^2 + 0^2 + 0 + 0 = 0^2 + 1^2 + 0 + 0 = 0^2 + (-1)^2 + 0 + 0 = 0^2 + 0^2 + 1 + 0 = 0^2 + 0^2 + 0 + 1

Amiram Eldar, Table of n, a(n) for n = 0..10001
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

Cf. A000203, A193553, A237593, A239052, A239053, A239931.

[DivisorSigma(1, 4*n+1): n in [0..60]]; // Vincenzo Librandi, Sep 18 2015

Table[DivisorSigma[1, 4 n + 1], {n, 0, 57}] (* Michael De Vlieger, Aug 31 2015 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^14/eta(x+A)^6/eta(x^4+A)^4, n))} /* Michael Somos, Jul 04 2006 */

a(n) = sigma(4n+1) where sigma(n) = A000203(n) is the sum of the divisors of n.
Euler transform of period 4 sequence [ 6, -8, 6, -4, ...]. - Michael Somos, Jul 04 2006
Expansion of q^(-1/4)eta^14(q^2)/(eta^6(q)eta^4(q^4)) in powers of q. - Michael Somos, Jul 04 2006
Expansion of psi(q)^2*phi(q)^2, i.e., convolution of A004018 and A008441 [Hirschhorn]. - R. J. Mathar, Mar 24 2011
Sum_{k=0..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A239052 Sum of divisors of 4*n-2.

Original entry on oeis.org

3, 12, 18, 24, 39, 36, 42, 72, 54, 60, 96, 72, 93, 120, 90, 96, 144, 144, 114, 168, 126, 132, 234, 144, 171, 216, 162, 216, 240, 180, 186, 312, 252, 204, 288, 216, 222, 372, 288, 240, 363, 252, 324, 360, 270, 336, 384, 360, 294, 468, 306, 312, 576

Offset: 1

Omar E. Pol, Mar 09 2014

Bisection of A062731 (odd part).
a(n) is also the total number of cells in the n-th branch of the second quadrant of the spiral formed by the parts of the symmetric representation of sigma(4n-2). For the quadrants 1, 3, 4 see A112610, A239053, A193553. The spiral has been obtained according to the following way: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A237270, see example.
We can find the spiral on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

			Illustration of initial terms:
------------------------------------------------------
.        Branches of the spiral
.        in the second quadrant             n    a(n)
------------------------------------------------------
.
.                  _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|         4     24
.                 | |
.             12 _| |
.               |_ _|  _ _ _ _ _ _
.         12 _ _|     |  _ _ _ _ _|         3     18
.      _ _ _| |    9 _| |
.     |  _ _ _|  9 _|_ _|
.     | |      _ _| |      _ _ _ _
.     | |     |  _ _| 12 _|  _ _ _|         2     12
.     | |     | |      _|   |
.     | |     | |     |  _ _|
.     | |     | |     | |    3 _ _
.     | |     | |     | |     |  _|         1      3
.     |_|     |_|     |_|     |_|
.
For n = 4 the sum of divisors of 4*n-2 is 1 + 2 + 7 + 14 = A000203(14) = 24. On the other hand the parts of the symmetric representation of sigma(14) are [12, 12] and the sum of them is 12 + 12 = 24, equaling the sum of divisors of 14, so a(4) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A008438, A016825, A062731, A074400, A112610, A193553, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239050, A239053, A244050, A245092, A262626.

a[n_] := DivisorSigma[1, 4*n - 2]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)

a(n) = A000203(4n-2) = A000203(A016825(n-1)).
a(n) = 3*A008438(n-1). - Joerg Arndt, Mar 09 2014
Sum_{k=1..n} a(k) = (3*Pi^2/8) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A240020 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(2n-1).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 3, 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 5, 5, 11, 12, 12, 13, 5, 13, 14, 6, 6, 14, 15, 15, 16, 16, 17, 7, 7, 17, 18, 12, 18, 19, 19, 20, 8, 8, 20, 21, 21, 22, 22, 23, 32, 23, 24, 24, 25, 7, 25, 26, 10, 10, 26, 27, 27, 28, 8, 8, 28, 29, 11, 11, 29, 30, 30, 31, 31, 32, 12, 26, 12, 32, 33, 9, 9, 33, 34, 34

Offset: 1

Omar E. Pol, Mar 31 2014

Row n lists the parts of the symmetric representation of A008438(n-1).
Also these are the parts from the odd-indexed rows of A237270.
Also these are the parts in the quadrants 1 and 3 of the spiral described in A239660, see example.
Row sums give A008438.
The length of row n is A237271(2n-1).
Both column 1 and the right border are equal to n.
Note that also the sequence can be represented in a quadrant.
We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

			1;
2, 2;
3, 3;
4, 4;
5, 3, 5;
6, 6;
7, 7;
8, 8, 8;
9, 9;
10, 10;
11, 5, 5, 11;
12, 12;
13, 5, 13;
14, 6, 6, 14;
15, 15;
16, 16;
17, 7, 7, 17;
18, 12, 18;
19, 19;
20, 8, 8, 20;
21, 21;
22, 22;
23, 32, 23;
24, 24;
25, 7, 25;
...
Illustration of initial terms (rows 1..8):
.
.                                   _ _ _ _ _ _ _ 7
.                                  |_ _ _ _ _ _ _|
.                                                |
.                                                |_ _
.                                   _ _ _ _ _ 5      |_
.                                  |_ _ _ _ _|         |
.                                            |_ _ 3    |_ _ _ 7
.                                            |_  |         | |
.                                   _ _ _ 3    |_|_ _ 5    | |
.                                  |_ _ _|         | |     | |
.                                        |_ _ 3    | |     | |
.                                          | |     | |     | |
.                                   _ 1    | |     | |     | |
.     _       _       _       _    |_|     |_|     |_|     |_|
.    | |     | |     | |     | |
.    | |     | |     | |     |_|_ _
.    | |     | |     | |    2  |_ _|
.    | |     | |     |_|_     2
.    | |     | |    4    |_
.    | |     |_|_ _        |_ _ _ _
.    | |    6      |_      |_ _ _ _|
.    |_|_ _ _        |_   4
.   8      | |_ _      |
.          |_    |     |_ _ _ _ _ _
.            |_  |_    |_ _ _ _ _ _|
.           8  |_ _|  6
.                  |
.                  |_ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _|
.                 8
.
The figure shows the quadrants 1 and 3 of the spiral described in A239660.
For n = 5 we have that 2*5 - 1 = 9 and the 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both symmetric Dyck paths there are three regions (or parts) of sizes [5, 3, 5], so row 5 is [5, 3, 5], see the third arm of the spiral in the first quadrant.
The sum of divisors of 9 is 1 + 3 + 9 = A000203(9) = 13. On the other hand the sum of the parts of the symmetric representation of sigma(9) is 5 + 3 + 5 = 13, equaling the sum of divisors of 9.

Cf. A000203, A005408, A008438, A112610, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A239053, A239660, A239931, A239933, A244050, A245092, A262626.

A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

Original entry on oeis.org

0, 1, 4, 8, 14, 17, 25, 26, 35, 36, 46, 43, 58, 54, 66, 62, 79, 73, 88, 77, 101, 94, 110, 92, 120, 115, 133, 113, 138, 126, 158, 134, 167, 143, 165, 150, 193, 177, 189, 154, 206, 188, 228, 182, 224, 206, 234, 198, 244, 229, 274, 222, 263, 224, 272, 246, 312, 272, 290, 230, 318, 290, 326, 262, 327, 315, 355, 296

Offset: 1

Omar E. Pol, Dec 24 2014

This is also a rectangular array read by rows, with four columns, in which T(j,k) is the number of cells (also the area) of the j-th gap between the arms in the k-th quadrant of the spiral of the symmetric representation of sigma described in A239660, with j >= 1 and 1 <= k <= 4 and starting with T(1,1) = 0, 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

			a(5) = sigma(4) + sigma(3) + sigma(2) = 7 + 4 + 3 = 14. On the other hand a(5) = A024916(4) - A024916(1) = 15 - 1 = 14.
...
Also, if written as a rectangular array T(j,k) with four columns the sequence begins:
    0,   1,   4,   8;
   14,  17,  25,  26;
   35,  36,  46,  43;
   58,  54,  66,  62;
   79,  73,  88,  77;
  101,  94, 110,  92;
  120, 115, 133, 113;
  138, 126, 158, 134;
  167, 143, 165, 150;
  193, 177, 189, 154;
  206, 188, 228, 182;
  224, 206, 234, 198;
  244, 229, 274, 222;
  263, 224, 272, 246;
  312, 272, 290, 230;
  318, 290, 326, 262;
  ...
In this case T(2,1) = a(5) = 14.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A010883, A024916, A092403, A112610, A193553, A196020, A236104, A237270, A237271, A237593, A239052, A239053, A239931-A239934, A239660, A240020, A244050, A245092, A262626.

L:= [0,0,0,seq(numtheory:-sigma(n), n=1..100)]:
L[1..101]+L[2..102]+L[3..103]; # Robert Israel, Dec 07 2016

a252922[n_] := Block[{f}, f[1] = 0; f[2] = 1; f[3] = 4;
  f[x_] := DivisorSigma[1, x - 1] + DivisorSigma[1, x - 2] +
DivisorSigma[1, x - 3]; Table[f[i], {i, n}]]; a252922[68] (* Michael De Vlieger, Dec 27 2014 *)

v=concat([0,1,4],vector(100,n,sigma(n)+sigma(n+1)+sigma(n+2))) \\ Derek Orr, Dec 30 2014

a(1) = 0, a(2) = sigma(1) = 1, a(3) = sigma(2) + sigma(1) = 4; for n >= 4, a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3).

a(n) = A024916(n-1) - A024916(n-4) for n >= 5.

cp's OEIS Frontend

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

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A239050 a(n) = 4*sigma(n).

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A239933 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-1).

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A112610 Number of representations of n as a sum of two squares and two triangular numbers.

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A239052 Sum of divisors of 4*n-2.

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A240020 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(2n-1).

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A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

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