A182181 -id:A182181 - cp's OEIS frontend

A182244 Sum of all parts of the shell model of partitions of A135010 with n regions.

1, 4, 9, 11, 20, 23, 35, 37, 43, 46, 66, 69, 76, 80, 105, 107, 113, 116, 129, 134, 138, 176, 179, 186, 190, 204, 207, 216, 221, 270, 272, 278, 281, 294, 299, 303, 326, 330, 340, 346, 351, 420, 423, 430, 434, 448, 451, 460, 465, 492, 497, 501, 516, 523, 529, 616

Offset: 1

Omar E. Pol, Apr 23 2012

			The first four regions of the shell model of partitions are [1],[2, 1],[3, 1, 1],[2], so a(4) = (1)+(2+1)+(3+1+1)+(2) = 11.
Written as a triangle begins:
1;
4;
9;
11,  20;
23,  35;
37,  43, 46, 66;
69,  76, 80,105;
107,113,116,129,134,138,176;
179,186,190,204,207,216,221,270;
272,278,281,294,299,303,326,330,340,346,351,420;
423,430,434,448,451,460,465,492,497,501,516,523,529,616;
...
From _Omar E. Pol_, Aug 08 2013: (Start)
Illustration of initial terms:
.                                                _ _ _ _ _
.                                      _ _ _    |_ _ _    |
.                            _ _ _ _  |_ _ _|_  |_ _ _|_  |
.                    _ _    |_ _    | |_ _    | |_ _    | |
.            _ _ _  |_ _|_  |_ _|_  | |_ _|_  | |_ _|_  | |
.      _ _  |_ _  | |_ _  | |_ _  | | |_ _  | | |_ _  | | |
.  _  |_  | |_  | | |_  | | |_  | | | |_  | | | |_  | | | |
. |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
.
.  1    4      9       11       20        23        35
.
.                                          _ _ _ _ _ _
.                             _ _ _       |_ _ _      |
.                _ _ _ _     |_ _ _|_     |_ _ _|_    |
.   _ _         |_ _    |    |_ _    |    |_ _    |   |
.  |_ _|_ _ _   |_ _|_ _|_   |_ _|_ _|_   |_ _|_ _|_  |
.  |_ _ _    |  |_ _ _    |  |_ _ _    |  |_ _ _    | |
.  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  | |
.  |_ _    | |  |_ _    | |  |_ _    | |  |_ _    | | |
.  |_ _|_  | |  |_ _|_  | |  |_ _|_  | |  |_ _|_  | | |
.  |_ _  | | |  |_ _  | | |  |_ _  | | |  |_ _  | | | |
.  |_  | | | |  |_  | | | |  |_  | | | |  |_  | | | | |
.  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|_|
.
.       37           43           46           66
(End)

Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A186412. Row j has length A187219(j). Right border gives A066186.

Cf. A000041, A135010, A138121, A141285, A182244, A182181, A182276, A186114, A206437, A210990.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A186412 = {}; l = {};
For[j = 1, j <= 56, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
  ];
Accumulate@A186412  (* Robert Price, Jul 25 2020 *)

a(A000041(k)) = A066186(k), k >= 1.

A210990 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 10, 21, 26, 44, 51, 75, 80, 92, 99, 136, 143, 157, 166, 213, 218, 230, 237, 260, 271, 280, 348, 355, 369, 378, 403, 410, 427, 438, 526, 531, 543, 550, 573, 584, 593, 631, 640, 659, 672, 683, 804, 811, 825, 834, 859, 866, 883, 894, 938, 949, 958

Offset: 0

Omar E. Pol, Apr 23 2012

nonn

Each part is represented by a cuboid of sides 1 X 1 X k where k is the size of the part. For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.   A182181(11) = 35            A182244(11) = 66
.
.   6                             * * * * * 6
.   3 3                      P    * * 3 * * 3
.   2   4                    a    * * * 4 * 2
.   2   2 2                  r    * 2 * 2 * 2
.   1       5                t    * * * * 5 1
.   1       2 3              i    * * 3 * 2 1
.   1       1   4            t    * * * 4 1 1
.   1       1   2 2          i    * 2 * 2 1 1
.   1       1   1   3        o    * * 3 1 1 1
.   1       1   1   1 2      n    * 2 1 1 1 1
.   1       1   1   1 1 1    s    1 1 1 1 1 1
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.                               A182727(11) = 35
.
The areas of the shadows of the three views are A182244(11) = 66, A182181(11) = 35 and A182727(11) = 35, therefore the total area of the three shadows is 66+35+35 = 136, so a(11) = 136.

Other versions: A207380, A210970, A210979, A210980, A210991.

Cf. A135010, A138121, A141285, A194446, A182181, A182727, A186114, A206437.

a(n) = A182244(n) + A182727(n) + A182181(n), n >= 1.

a(A000041(n)) = 2*A006128(n) + A066186(n).

A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 9, 18, 21, 35, 39, 58, 61, 67, 71, 99, 103, 110, 115, 152, 155, 161, 165, 175, 181, 186, 238, 242, 249, 254, 265, 269, 277, 283, 352, 355, 361, 365, 375, 381, 386, 401, 406, 415, 422, 428, 522, 526, 533, 538, 549, 553, 561, 567, 584, 590, 595, 606

Offset: 0

Omar E. Pol, Apr 30 2012

nonn

It appears that if n is a partition number A000041 then the rotated structure with n regions shows each row as a partition of k such that A000041(k) = n (see example).

For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.     A182181(11) = 35           A210692(11) = 29
.
.   1                                       1
.   1                                       1
.   1                                       1
.   1                                       1
.   1       1                             1 1
.   1       1                             1 1
.   1       1   1                       1 1 1
.   2       1   1                       1 1 2
.   2       1   1   1                 1 1 1 2
.   3   2   2   2   1 1             1 1 2 2 3
.   6 3 4 2 5 3 4 2 3 2 1         1 2 3 4 5 6
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.
.                                A182727(11) = 35
.
The areas of the shadows of the three views are A182181(11) = 35, A182727(11) = 35 and A210692(11) = 29, therefore the total area of the three shadows is 35+35+29 = 99, so a(11) = 99.
Since n = 11 is a partition number A000041 we can see that the rotated structure with 11 regions shows each row as a partition of 6 because A000041(6) = 11. See below:
.
.                      6
.                    3   3
.                  4       2
.                2   2       2
.              5               1
.            3   2               1
.          4       1               1
.        2   2       1               1
.      3       1       1               1
.    2   1       1       1               1
.  1   1   1       1       1               1
.

Other versions: A207380, A210970, A210979, A210980, A210990.

Cf. A000041, A026905, A135010, A138121, A141285, A182703, A194446, A182181, A182727, A186114, A206437, A210692.

a(n) = A182181(n) + A182727(n) + A210692(n).

a(A000041(n)) = 2*A006128(n) + A026905(n).

A299474 a(n) = 4*p(n), where p(n) is the number of partitions of n.

Original entry on oeis.org

4, 4, 8, 12, 20, 28, 44, 60, 88, 120, 168, 224, 308, 404, 540, 704, 924, 1188, 1540, 1960, 2508, 3168, 4008, 5020, 6300, 7832, 9744, 12040, 14872, 18260, 22416, 27368, 33396, 40572, 49240, 59532, 71908, 86548, 104060, 124740, 149352, 178332, 212696, 253044, 300700, 356536, 422232, 499016, 589092, 694100, 816904

Offset: 0

Omar E. Pol, Feb 10 2018

nonn

For n >= 1, a(n) is also the number of edges in the diagram of partitions of n, in which A299475(n) is the number of vertices and A000041(n) is the number of regions (see example and Euler's formula).

			Construction of a modular table of partitions in which a(n) is the number of edges of the diagram after n-th stage (n = 1..6):
--------------------------------------------------------------------------------
n ........:   1     2       3         4           5             6     (stage)
a(n)......:   4     8      12        20          28            44     (edges)
A299475(n):   4     7      10        16          22            34     (vertices)
A000041(n):   1     2       3         5           7            11     (regions)
--------------------------------------------------------------------------------
r     p(n)
--------------------------------------------------------------------------------
.             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _
1 .... 1 ....|_|  |_| |  |_| | |  |_| | | |  |_| | | | |  |_| | | | | |
2 .... 2 .........|_ _|  |_ _| |  |_ _| | |  |_ _| | | |  |_ _| | | | |
3 .... 3 ................|_ _ _|  |_ _ _| |  |_ _ _| | |  |_ _ _| | | |
4                                 |_ _|   |  |_ _|   | |  |_ _|   | | |
5 .... 5 .........................|_ _ _ _|  |_ _ _ _| |  |_ _ _ _| | |
6                                            |_ _ _|   |  |_ _ _|   | |
7 .... 7 ....................................|_ _ _ _ _|  |_ _ _ _ _| |
8                                                         |_ _|   |   |
9                                                         |_ _ _ _|   |
10                                                        |_ _ _|     |
11 .. 11 .................................................|_ _ _ _ _ _|
.
Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.
Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.
The total number of parts equals the sum of largest parts.
Note that every diagram contains all previous diagrams.
An infinite diagram is a table of all partitions of all positive integers.

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

k times partition numbers: A000041 (k=1), A139582 (k=2), A299473 (k=3), this sequence (k=4).

Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.

List([0..50],n->4*NrPartitions(n)); # Muniru A Asiru, Jul 10 2018

with(combinat): seq(4*numbpart(n),n=0..50); # Muniru A Asiru, Jul 10 2018

4*PartitionsP[Range[0,50]] (* Harvey P. Dale, Dec 05 2023 *)

a(n) = 4*numbpart(n); \\ Michel Marcus, Jul 15 2018

from sympy.ntheory import npartitions
def a(n): return 4*npartitions(n)
print([a(n) for n in range(51)]) # Michael S. Branicky, Apr 04 2021

a(n) = 4*A000041(n) = 2*A139582(n).
a(n) = A000041(n) + A299475(n) - 1, n >= 1 (Euler's formula).
a(n) = A000041(n) + A299473(n). - Omar E. Pol, Aug 11 2018

A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

Original entry on oeis.org

1, 4, 7, 10, 16, 22, 34, 46, 67, 91, 127, 169, 232, 304, 406, 529, 694, 892, 1156, 1471, 1882, 2377, 3007, 3766, 4726, 5875, 7309, 9031, 11155, 13696, 16813, 20527, 25048, 30430, 36931, 44650, 53932, 64912, 78046, 93556, 112015, 133750, 159523, 189784, 225526, 267403, 316675, 374263, 441820, 520576, 612679

Offset: 0

Omar E. Pol, Feb 11 2018

nonn

For n >= 1, A299474(n) is the number of edges and A000041(n) is the number of regions in the mentioned diagram (see example and Euler's formula).

			Construction of a modular table of partitions in which a(n) is the number of vertices of the diagram after n-th stage (n = 1..6):
--------------------------------------------------------------------------------
n ........:   1     2       3         4           5             6     (stage)
a(n)......:   4     7      10        16          22            34     (vertices)
A299474(n):   4     8      12        20          28            44     (edges)
A000041(n):   1     2       3         5           7            11     (regions)
--------------------------------------------------------------------------------
r     p(n)
--------------------------------------------------------------------------------
.             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _
1 .... 1 ....|_|  |_| |  |_| | |  |_| | | |  |_| | | | |  |_| | | | | |
2 .... 2 .........|_ _|  |_ _| |  |_ _| | |  |_ _| | | |  |_ _| | | | |
3 .... 3 ................|_ _ _|  |_ _ _| |  |_ _ _| | |  |_ _ _| | | |
4                                 |_ _|   |  |_ _|   | |  |_ _|   | | |
5 .... 5 .........................|_ _ _ _|  |_ _ _ _| |  |_ _ _ _| | |
6                                            |_ _ _|   |  |_ _ _|   | |
7 .... 7 ....................................|_ _ _ _ _|  |_ _ _ _ _| |
8                                                         |_ _|   |   |
9                                                         |_ _ _ _|   |
10                                                        |_ _ _|     |
11 .. 11 .................................................|_ _ _ _ _ _|
.
Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.
Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.
The total number of parts equals the sum of largest parts.
Note that every diagram contains all previous diagrams.
An infinite diagram is a table of all partitions of all positive integers.

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

Cf. A000041, A135010, A139582, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299474.

a(n) = if (n==0, 1, 1+3*numbpart(n)); \\ Michel Marcus, Jul 15 2018

a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.

a(n) = A299474(n) - A000041(n) + 1, n >= 1 (Euler's formula).

A182727 Sum of largest parts of the shell model of partitions with n regions.

Original entry on oeis.org

1, 3, 6, 8, 12, 15, 20, 22, 26, 29, 35, 38, 43, 47, 54, 56, 60, 63, 69, 74, 78, 86, 89, 94, 98, 105, 108, 114, 119, 128, 130, 134, 137, 143, 148, 152, 160, 164, 171, 177, 182, 192, 195, 200, 204, 211, 214, 220, 225, 234, 239, 243, 251, 258, 264, 275, 277, 281

Offset: 1

Omar E. Pol, Jan 25 2011

Question: Is there some connection with fractals?

			For n = 6 the largest parts of the first six regions of the shell model of partitions are 1, 2, 3, 2, 4, 3, so a(6) = 1+2+3+2+4+3 = 15.
Written as a triangle begins:
1;
3;
6;
8,   12;
15,  20;
22,  26, 29, 35;
38,  43, 47, 54;
56,  60, 63, 69, 74, 78, 86;
89,  94, 98,105,108,114,119,128;
130,134,137,143,148,152,160,164,171,177,182,192;
195,200,204,211,214,220,225,234,239,243,251,258,264,275;

Alois P. Heinz, Table of n, a(n) for n = 1..4000
Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A141285. Row j has length A187219(j). Right border gives A006128.

Cf. A135010, A141285, A182728, A182729, A186114, A206437, A182181, A182244, A210990.

a(A000041(n)) = A182181(A000041(n)) = A006128(n). - Omar E. Pol, May 24 2012

New name from Omar E. Pol, Apr 26 2012

A182276 Sum of all parts minus the total number of parts of the shell model of partitions with n regions.

Original entry on oeis.org

0, 1, 3, 4, 8, 10, 15, 16, 20, 22, 31, 33, 38, 41, 51, 52, 56, 58, 67, 71, 74, 90, 92, 97, 100, 110, 112, 119, 123, 142, 143, 147, 149, 158, 162, 165, 181, 184, 192, 197, 201, 228, 230, 235, 238, 248, 250, 257, 261, 280, 284, 287, 299, 305, 310, 341

Offset: 1

Omar E. Pol, Apr 23 2012

nonn

For the definition of "region of n" see A206437.

			Written has a triangle:
0,
1,
3,
4,    8;
10,  15;
16,  20, 22, 31;
33,  38, 41, 51;
52,  56, 58, 67, 71, 74, 90;
92,  97,100,110,112,119,123,142;
143,147,149,158,162,165,181,184,192,197,201,228;
230,235,238,248,250,257,261,280,284,287,299,305,310,341;

Row j has length A187219(j). Right border gives A196087.

Cf. A000041, A066186, A006128, A135010, A138121, A182181, A182244, A186114, A206437, A207035.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
reg = {}; l = {};
For[j = 1, j <= 56, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  t = Take[Reverse[First /@ lex[mx]], j - i];
  AppendTo[reg, Total@t - Length@t]
  ];
Accumulate@reg  (* Robert Price, Jul 25 2020 *)

a(n) = A182244(n) - A182181(n).

a(A000041(n)) = A196087(n).

A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n.

Original entry on oeis.org

3, 3, 6, 9, 15, 21, 33, 45, 66, 90, 126, 168, 231, 303, 405, 528, 693, 891, 1155, 1470, 1881, 2376, 3006, 3765, 4725, 5874, 7308, 9030, 11154, 13695, 16812, 20526, 25047, 30429, 36930, 44649, 53931, 64911, 78045, 93555, 112014, 133749, 159522, 189783, 225525, 267402, 316674, 374262, 441819, 520575, 612678

Offset: 0

Omar E. Pol, Feb 10 2018

nonn

For n >= 1, a(n) is also the number of vertices in the minimalist diagram of partitions of n, in which A139582(n) is the number of line segments and A000041(n) is the number of open regions (see example).

			Construction of a minimalist version of a modular table of partitions in which a(n) is the number of vertices of the diagram after n-th stage (n = 1..6):
-----------------------------------------------------------------------------------
n.........:    1     2       3         4           5           6   (stage)
A000041(n):    1     2       3         5           7          11   (open regions)
A139582(n):    2     4       6        10          14          22   (line segments)
a(n)......:    3     6       9        15          21          33   (vertices)
-----------------------------------------------------------------------------------
r     p(n)
-----------------------------------------------------------------------------------
.
1 .... 1 .... _|   _| |   _| | |   _| | | |   _| | | | |   _| | | | | |
2 .... 2 ......... _ _|   _ _| |   _ _| | |   _ _| | | |   _ _| | | | |
3 .... 3 ................ _ _ _|   _ _ _| |   _ _ _| | |   _ _ _| | | |
4                                  _ _|   |   _ _|   | |   _ _|   | | |
5 .... 5 ......................... _ _ _ _|   _ _ _ _| |   _ _ _ _| | |
6                                             _ _ _|   |   _ _ _|   | |
7 .... 7 .................................... _ _ _ _ _|   _ _ _ _ _| |
8                                                          _ _|   |   |
9                                                          _ _ _ _|   |
10                                                         _ _ _|     |
11 .. 11 ................................................. _ _ _ _ _ _|
.
The r-th horizontal line segment has length A141285(r).
The r-th vertical line segment has length A194446(r).
An infinite diagram is a minimalist table of all partitions of all positive integers.

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

k times partition numbers: A000041 (k=1), A139582 (k=2), this sequence (k=3), A299474 (k=4).

Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.

a(n) = 3*A000041(n) = A000041(n) + A139582(n).

a(n) = A299475(n) - 1, n >= 1.

A220487 Partial sums of triangle A206437.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 11, 15, 17, 18, 19, 20, 23, 28, 30, 31, 32, 33, 34, 35, 37, 41, 43, 46, 52, 55, 57, 59, 60, 61, 62, 63, 64, 65, 66, 69, 74, 76, 80, 87, 90, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 111, 113, 116, 122, 125, 127, 129, 134, 138

Offset: 1

Omar E. Pol, Jan 18 2013

			When written as an irregular triangle in which row j has length A194446(j) then the right border gives A182244. Also the records of row lengths give the partition numbers (A000041) of the positive integers as shown below:
1;
3, 4;
7, 8, 9;
11;
15,17,18,19,20;
23;
28,30,31,32,33,34,35;
37;
41,43;
46;
52,55,57,59,60,61,62,63,64,65,66;
69;
74,76;
80;
87,90,92,94,95,96,97,98,99,100,101,102,103,104,105;
...
Also when written as an irregular triangle in which row j has length A138137(j) then the right border gives A066186 as shown below:
1;
3, 4;
7, 8, 9;
11,15,17,18,19,20;
23,28,30,31,32,33,34,35;
37,41,43,46,52,55,57,59,60,61,62,63,64,65,66;
69,74,76,80,87,90,92,94,95,96,97,98,99,100,101,102,103,104,105;
...

Cf. A000041, A006128, A066186, A135010, A138137, A138879, A141285, A182181, A182244, A186412, A194446, A206437.

a(A182181(n)) = A182244(n), n >= 1.

a(A006128(n)) = A066186(n), n >= 1.

cp's OEIS Frontend

A182244 Sum of all parts of the shell model of partitions of A135010 with n regions.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A210990 Total area of the shadows of the three views of the shell model of partitions with n regions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A299474 a(n) = 4*p(n), where p(n) is the number of partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Python

Formula

A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A182727 Sum of largest parts of the shell model of partitions with n regions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A182276 Sum of all parts minus the total number of parts of the shell model of partitions with n regions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n.

Views

Author

Keywords

Comments

Examples

Links