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 1-10 of 14 results. Next

A225620 Indices of partitions in the table of compositions of A228351.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 100, 104, 106, 112, 116, 120, 122, 124, 126, 127, 128, 136, 144, 160, 164, 168, 170, 192, 200, 208, 212, 224, 228, 232, 234, 240, 244, 248, 250, 252, 254, 255
Offset: 1

Views

Author

Omar E. Pol, Aug 03 2013

Keywords

Comments

Also triangle read by rows in which T(n,k) is the decimal representation of a binary number whose mirror represents the k-th partition of n according with the list of juxtaposed reverse-lexicographically ordered partitions of the positive integers (A026792).
In order to construct this sequence as a triangle we use the following rules:
- In the list of A026792 we replace each part of size j of the k-th partition of n by concatenation of j - 1 zeros and only one 1.
- Then replace this new set of parts by the concatenation of its parts.
- Then replace this string by its mirror version which is a binary number.
T(n,k) is the decimal value of this binary number, which represents the k-th partition of n (see example).
The partitions of n are represented by a subsequence with A000041(n) integers starting with 2^(n-1) and ending with 2^n - 1, n >= 1. The odd numbers of the sequence are in A000225.
First differs from A065609 at a(23).
Conjecture: this sequence is a sorted version of b(n) where b(2^k) = 2^k for k >= 0, b(n) = A080100(n)*(2*b(A053645(n)) + 1) otherwise. - Mikhail Kurkov, Oct 21 2023

Examples

			T(6,8) = 58 because 58 in base 2 is 111010 whose mirror is 010111 which is the concatenation of 01, 01, 1, 1, whose number of digits are 2, 2, 1, 1, which are also the 8th partition of 6.
Illustration of initial terms:
The sequence represents a table of partitions (see below):
--------------------------------------------------------
.            Binary                        Partitions
n  k  T(n,k) number  Mirror   Diagram       (A026792)
.                                          1 2 3 4 5 6
--------------------------------------------------------
.                             _
1  1     1       1    1        |           1,
.                             _ _
1  1     2      10    01      _  |           2,
2  2     3      11    11       | |         1,1,
.                             _ _ _
3  1     4     100    001     _ _  |           3,
3  2     6     110    011     _  | |         2,1,
3  3     7     111    111      | | |       1,1,1,
.                             _ _ _ _
4  1     8    1000    0001    _ _    |           4,
4  2    10    1010    0101    _ _|_  |         2,2,
4  3    12    1100    0011    _ _  | |         3,1,
4  4    14    1110    0111    _  | | |       2,1,1,
4  5    15    1111    1111     | | | |     1,1,1,1,
.                             _ _ _ _ _
5  1    16   10000    00001   _ _ _    |           5,
5  2    20   10100    00101   _ _ _|_  |         3,2,
5  3    24   11000    00011   _ _    | |         4,1,
5  4    26   11010    01011   _ _|_  | |       2,2,1,
5  5    28   11100    00111   _ _  | | |       3,1,1,
5  6    30   11110    01111   _  | | | |     2,1,1,1,
5  7    31   11111    11111    | | | | |   1,1,1,1,1,
.                             _ _ _ _ _ _
6  1    32  100000    000001  _ _ _      |           6
6  2    36  100100    001001  _ _ _|_    |         3,3,
6  3    40  101000    000101  _ _    |   |         4,2,
6  4    42  101010    010101  _ _|_ _|_  |       2,2,2,
6  5    48  110000    000011  _ _ _    | |         5,1,
6  6    52  110100    001011  _ _ _|_  | |       3,2,1,
6  7    56  111000    000111  _ _    | | |       4,1,1,
6  8    58  111010    010111  _ _|_  | | |     2,2,1,1,
6  9    60  111100    001111  _ _  | | | |     3,1,1,1,
6  10   62  111110    011111  _  | | | | |   2,1,1,1,1,
6  11   63  111111    111111   | | | | | | 1,1,1,1,1,1,
.
Triangle begins:
  1;
  2,   3;
  4,   6,  7;
  8,  10, 12, 14, 15;
  16, 20, 24, 26, 28, 30, 31;
  32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63;
  ...
From _Gus Wiseman_, Apr 01 2020: (Start)
Using the encoding of A066099, this sequence ranks all finite nonempty multisets, as follows.
   1: {1}
   2: {2}
   3: {1,1}
   4: {3}
   6: {1,2}
   7: {1,1,1}
   8: {4}
  10: {2,2}
  12: {1,3}
  14: {1,1,2}
  15: {1,1,1,1}
  16: {5}
  20: {2,3}
  24: {1,4}
  26: {1,2,2}
  28: {1,1,3}
  30: {1,1,1,2}
  31: {1,1,1,1,1}
(End)

Crossrefs

Column 1 is A000079. Row n has length A000041(n). Right border gives A000225.
Cf. A026792, A065609, A135010, A141285, A186114, A194446, A194546, A206437, A207779, A211978, A225600, A225610, A228351.
The case covering an initial interval is A333379 or A333380.
All of the following pertain to compositions in the order of A066099.
- The weakly increasing version is this sequence.
- The weakly decreasing version is A114994.
- The strictly increasing version is A333255.
- The strictly decreasing version is A333256.
- The unequal version is A233564.
- The equal version is A272919.
- The case covering an initial interval is A333217.
- Initial intervals are ranked by A164894.
- Reversed initial intervals are ranked by A246534.
Cf. A000120, A029931, A048793, A070939, A124766, A233249, A333219, A333220.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],LessEqual@@stc[#]&] (* Gus Wiseman, Apr 01 2020 *)

Formula

Conjecture: a(A000070(m) - k) = 2^m - A228354(k) for m > 0, 0 < k <= A000041(m). - Mikhail Kurkov, Oct 20 2023

A225600 Toothpick sequence related to integer partitions (see Comments lines for definition).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 14, 15, 19, 24, 27, 28, 33, 40, 42, 43, 47, 49, 52, 53, 59, 70, 73, 74, 79, 81, 85, 86, 93, 108, 110, 111, 115, 117, 120, 121, 127, 131, 136, 137, 141, 142, 150, 172, 175, 176, 181, 183, 187, 188, 195, 199, 202, 203, 209, 211, 216, 217, 226, 256
Offset: 0

Views

Author

Omar E. Pol, Jul 28 2013

Keywords

Comments

This infinite toothpick structure is a minimalist diagram of regions of the set of partitions of all positive integers. For the definition of "region" see A206437. The sequence shows the growth of the diagram as a cellular automaton in which the "input" is A141285 and the "output” is A194446.
To define the sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks.
If n is odd we place A141285((n+1)/2) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, (n+1)/2).
If n is even we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. In this case the number of toothpicks added in vertical direction is equal to A194446(n/2).
The sequence gives the number of toothpicks after n stages. A220517 (the first differences) gives the number of toothpicks added at the n-th stage.
Also the toothpick structure (HV/HHVV/HHHVVV/HHV/HHHHVVVVV...) can be transformed in a Dyck path (UDUUDDUUUDDDUUDUUUUDDDDD...) in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps, so the sequence can be represented by the vertices (or the number of steps from the origin) of the Dyck path. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See Example section. See also A211978, A220517, A225610.

Examples

			For n = 30 the structure has 108 toothpicks, so a(30) = 108.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15  _ _ _ _      |
4 + 3                               _ _ _ _|_    |
5 + 2                               _ _ _    |   |
3 + 2 + 2                           _ _ _|_ _|_  |
6 + 1                           11  _ _ _      | |
3 + 3 + 1                           _ _ _|_    | |
4 + 2 + 1                           _ _    |   | |
2 + 2 + 2 + 1                       _ _|_ _|_  | |
5 + 1 + 1                        7  _ _ _    | | |
3 + 2 + 1 + 1                       _ _ _|_  | | |
4 + 1 + 1 + 1                    5  _ _    | | | |
2 + 2 + 1 + 1 + 1                   _ _|_  | | | |
3 + 1 + 1 + 1 + 1                3  _ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2  _  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1   | | | | | | |
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms:
.
.                              _ _ _    _ _ _
.                _ _   _ _     _ _      _ _  |
.      _    _    _     _  |    _  |     _  | |
.            |    |     | |     | |      | | |
.
.      1    2     4     6       9        12
.
.
.                          _ _ _ _     _ _ _ _
.      _ _       _ _       _ _         _ _    |
.      _ _ _     _ _|_     _ _|_       _ _|_  |
.      _ _  |    _ _  |    _ _  |      _ _  | |
.      _  | |    _  | |    _  | |      _  | | |
.       | | |     | | |     | | |       | | | |
.
.        14        15         19          24
.
.
.                          _ _ _ _ _    _ _ _ _ _
.    _ _ _      _ _ _      _ _ _        _ _ _    |
.    _ _ _ _    _ _ _|_    _ _ _|_      _ _ _|_  |
.    _ _    |   _ _    |   _ _    |     _ _    | |
.    _ _|_  |   _ _|_  |   _ _|_  |     _ _|_  | |
.    _ _  | |   _ _  | |   _ _  | |     _ _  | | |
.    _  | | |   _  | | |   _  | | |     _  | | | |
.     | | | |    | | | |    | | | |      | | | | |
.
.       27         28         33            40
.
Illustration of initial terms as vertices (or the number of steps from the origin) of a Dyck path:
.
7                                    33
.                                    /\
5                      19           /  \
.                      /\          /    \
3            9        /  \     27 /      \
2       4    /\   14 /    \    /\/        \
1    1  /\  /  \  /\/      \  / 28         \
.    /\/  \/    \/ 15       \/              \
.   0  2   6    12          24              40
.

Links

Crossrefs

Cf. A000041, A006128, A135010, A138137, A139250, A139582, A141285, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225610.

Formula

a(A139582(n)) = a(2*A000041(n)) = 2*A006128(n) = A211978(n), n >= 1.

A225610 Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.

Original entry on oeis.org

1, 4, 10, 18, 33, 52, 87, 130, 202, 295, 436, 617, 887, 1226, 1709, 2327, 3173, 4244, 5691, 7505, 9907, 12917, 16822, 21690, 27947, 35685, 45506, 57625, 72836, 91500, 114760, 143143, 178235, 220908, 273268, 336670, 414041, 507298, 620455, 756398, 920470
Offset: 0

Views

Author

Omar E. Pol, Jul 29 2013

Keywords

Comments

a(n) is also the total number of toothpicks in a toothpick structure which represents a diagram of regions of the set of partitions of n, n >= 1. The number of horizontal toothpicks is A225596(n). The number of vertical toothpicks is A093694(n). The difference between vertical toothpicks and horizontal toothpicks is A000041(n) - n = A000094(n+1). The total area (or total number of cells) of the diagram is A066186(n). The number of parts in the k-th region is A194446(k). The area (or number of cells) of the k-th region is A186412(k). For the definition of "region" see A206437. For a minimalist version of the diagram (which can be transformed into a Dyck path) see A211978. See also A225600.

Examples

			For n = 7 the total number of parts in all partitions of 7 plus the sum of largest parts in all partitions of 7 plus the number of partitions of 7 plus 7 is equal to A006128(7) + A006128(7) + A000041(7) + 7 = 54 + 54 + 15 + 7 = 130. On the other hand the number of toothpicks in the diagram of regions of the set of partitions of 7 is equal to 130, so a(7) = 130.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15 |_ _ _ _      |
4 + 3                              |_ _ _ _|_    |
5 + 2                              |_ _ _    |   |
3 + 2 + 2                          |_ _ _|_ _|_  |
6 + 1                           11 |_ _ _      | |
3 + 3 + 1                          |_ _ _|_    | |
4 + 2 + 1                          |_ _    |   | |
2 + 2 + 2 + 1                      |_ _|_ _|_  | |
5 + 1 + 1                        7 |_ _ _    | | |
3 + 2 + 1 + 1                      |_ _ _|_  | | |
4 + 1 + 1 + 1                    5 |_ _    | | | |
2 + 2 + 1 + 1 + 1                  |_ _|_  | | | |
3 + 1 + 1 + 1 + 1                3 |_ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2 |_  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1 |_|_|_|_|_|_|_|
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms as the number of toothpicks in a diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                        |_ _ _      |
.                                        |_ _ _|_    |
.                                        |_ _    |   |
.                             _ _ _ _ _  |_ _|_ _|_  |
.                            |_ _ _    | |_ _ _    | |
.                   _ _ _ _  |_ _ _|_  | |_ _ _|_  | |
.                  |_ _    | |_ _    | | |_ _    | | |
.           _ _ _  |_ _|_  | |_ _|_  | | |_ _|_  | | |
.     _ _  |_ _  | |_ _  | | |_ _  | | | |_ _  | | | |
. _  |_  | |_  | | |_  | | | |_  | | | | |_  | | | | |
.|_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 4    10     18       33         52          87

Links

Crossrefs

Cf. A000041, A000094, A006128, A066186, A093694, A133041, A135010, A138137, A139250, A139582, A141285, A182377, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225596, A225600.

Formula

a(n) = 2*A006128(n) + A000041(n) + n = A211978(n) + A133041(n) = A093694(n) + A006128(n) + n = A093694(n) + A225596(n).

A133041 Sum of n and partition number of n.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 17, 22, 30, 39, 52, 67, 89, 114, 149, 191, 247, 314, 403, 509, 647, 813, 1024, 1278, 1599, 1983, 2462, 3037, 3746, 4594, 5634, 6873, 8381, 10176, 12344, 14918, 18013, 21674, 26053, 31224, 37378, 44624, 53216
Offset: 0

Views

Author

Omar E. Pol, Oct 29 2007

Keywords

Links

Crossrefs

Cf. A001365, A014688, A064985, A090631, A102379.
Cf. A000041 (the partition numbers).
Partial sums of A232697.

Programs

Formula

a(n) = n + partition(n) = n + A000041(n).
a(n) = A207779(A000041(n)), n >= 1. - Omar E. Pol, Jul 17 2014

A228367 n-th element of the ruler function plus the highest power of 2 dividing n.

Original entry on oeis.org

2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 71, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7
Offset: 1

Views

Author

Omar E. Pol, Aug 22 2013

Keywords

Comments

a(n) is also the length of the n-th pair of orthogonal line segments in a diagram of compositions, see example.
a(n) is also the largest part plus the number of parts of the n-th region of the mentioned diagram (if the axes both "x" and "y" are included in the diagram).
a(n) is also the number of toothpicks added at n-th stage to the structure of A228366. Essentially the first differences of A228366.
The equivalent sequence for partitions is A207779.

Examples

			Illustration of initial terms (n = 1..16) using a diagram of compositions in which A001511(n) is the length of the horizontal line segment in row n and A006519(n) is the length of the vertical line segment ending in row n. Hence a(n) is the length of the n-th pair of orthogonal line segments. Also counting both the x-axis and the y-axis we have that A001511(n) is also the largest part of the n-th region of the diagram and A006519(n) is also the number of parts of the n-th region of the diagram, see below.
---------------------------------------------------------
.                Diagram of
n   A001511(n)  compositions   A006519(n)    a(n)
---------------------------------------------------------
1       1        _| | | | |        1          2
2       2        _ _| | | |        2          4
3       1        _|   | | |        1          2
4       3        _ _ _| | |        4          7
5       1        _| |   | |        1          2
6       2        _ _|   | |        2          4
7       1        _|     | |        1          2
8       4        _ _ _ _| |        8         12
9       1        _| | |   |        1          2
10      2        _ _| |   |        2          4
11      1        _|   |   |        1          2
12      3        _ _ _|   |        4          7
13      1        _| |     |        1          2
14      2        _ _|     |        2          4
15      1        _|       |        1          2
16      5        _ _ _ _ _|       16         21
...
If written as an irregular triangle the sequence begins:
  2;
  4;
  2, 7;
  2, 4, 2, 12;
  2, 4, 2, 7, 2, 4, 2, 21;
  2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38;
  ...
Row lengths is A011782. Right border gives A005126.
Counting both the x-axis and the y-axis we have that A038712(n) is the area (or the number of cells) of the n-th region of the diagram. Note that adding only the x-axis to the diagram we have a tree. - _Omar E. Pol_, Nov 07 2018

Links

Crossrefs

Cf. A001511, A001792, A005126, A006519, A011782, A038712, A139250, A139251, A207779, A228366.

Programs

Formula

a(n) = A001511(n) + A006519(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

Views

Author

Omar E. Pol, Feb 10 2018

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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.

Programs

  • GAP
    List([0..50],n->4*NrPartitions(n)); # Muniru A Asiru, Jul 10 2018
  • Maple
    with(combinat): seq(4*numbpart(n),n=0..50); # Muniru A Asiru, Jul 10 2018
  • Mathematica
    4*PartitionsP[Range[0,50]] (* Harvey P. Dale, Dec 05 2023 *)
  • PARI
    a(n) = 4*numbpart(n); \\ Michel Marcus, Jul 15 2018
  • Python
    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

Formula

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

Views

Author

Omar E. Pol, Feb 11 2018

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.
a(n) = A299474(n) - A000041(n) + 1, n >= 1 (Euler's formula).

A228109 Height after n-th step of an infinite staircase which is the lower part of a structure whose upper part is the infinite Dyck path of A228110.

Original entry on oeis.org

0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 1, 0, -1, 0, -1, 0, 1, 2, 1, 2, 3, 4, 3, 4, 3, 2, 1, 0, -1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 5, 4, 5, 4, 3, 2, 1, 0, -1
Offset: 0

Views

Author

Omar E. Pol, Aug 13 2013

Keywords

Comments

The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 5, the diagram 1 represents the partitions of 5. The diagram 2 shows separately the boundary segments southwest sides of the first seven regions of the diagram 1, see below:
.
j Diagram 1 Diagram 2
_
7 | _ | | _
6 | _| | | _ |
5 | | | | |
4 | |_ | | | |_ |
3 | | | | | | |
2 | | | | | | | | |
1 |||_||| | | | | |_
.
. 1 2 3 4 5
.
a(n) is the height after n-th step of an infinite staircase which is the lower part of a diagram of regions of the set of partitions of all positive integers. The upper part of the diagram is the infinite Dyck path mentioned in A228110. The diagram shows the shape of the successive regions of the set of partitions of all positive integers. The area of the n-th region is A186412(n).
For the height of the peaks and the valleys in the infinite Dyck path see A229946.

Examples

			Illustration of initial terms (n = 1..53):
5
4                                                      /
3                                 /\/\                /
2                                /    \            /\/
1                   /\/\      /\/      \        /\/
0          /\    /\/    \    /          \    /\/
-1 \/\/\/\/  \/\/        \/\/            \/\/
-2
The diagram shows the Dyck pack mentioned in A228110 together with the staircase illustrated above. The area of the n-th region is equal to A186412(n).
.
7...................................
.                                  /\
5.....................            /  \                /\
.                    /\          /    \          /\  / /
3...........        /  \        / /\/\ \        /  \/ /
2......    /\      /    \    /\/ /    \ \      /   /\/
1...  /\  /  \  /\/ /\/\ \  / /\/      \ \  /\/ /\/
0  /\/  \/ /\ \/ /\/    \ \/ /          \ \/ /\/
-1 \/\/\/\/  \/\/        \/\/            \/\/
.
Region:
.   1  2    3   4     5      6      7       8    9   10

Crossrefs

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228110, A229946.

A228110 Height after n-th step of the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, n >= 0, k >= 1.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0
Offset: 0

Views

Author

Omar E. Pol, Aug 10 2013

Keywords

Comments

The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y]. See below:
.
. j Diagram 1 Partitions Diagram 2
. _ _ _ _
. 15 | _ | 7 _ |
. 14 | _ | | 4+3 _ | |
. 13 | _ | | 5+2 _ | |
. 12 | _| |_ | 3+2+2 _| |_ |
. 11 | _ | | 6+1 _ | |
. 10 | _| | | 3+3+1 _ | | |
. 9 | | | | 4+2+1 | | |
. 8 | |_ | | | 2+2+2+1 |_ | | |
. 7 | _ | | | 5+1+1 _ | | |
. 6 | _| | | | 3+2+1+1 _ | | | |
. 5 | | | | | 4+1+1+1 | | | |
. 4 | |_ | | | | 2+2+1+1+1 |_ | | | |
. 3 | | | | | | 3+1+1+1+1 | | | | |
. 2 | | | | | | | 2+1+1+1+1+1 | | | | | |
. 1 |||_|||_|_| 1+1+1+1+1+1+1 | | | | | | |
.
. 1 2 3 4 5 6 7
.
The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1.
Also the diagram has the property that it can be transformed in a Dyck path (see example).
The sequence gives the height of the infinite Dyck path after n-th step.
The absolute values of the first differences give A000012.
For the height of the peaks and the valleys in the infinite Dyck path see A229946.
Q: Is this infinite Dyck path a fractal?

Examples

			Illustration of initial terms (n = 1..59):
.
11 ...........................................................
.                                                            /
.                                                           /
.                                                          /
7 ..................................                      /
.                                  /\                    /
5 ....................            /  \                /\/
.                    /\          /    \          /\  /
3 ..........        /  \        /      \        /  \/
2 .....    /\      /    \    /\/        \      /
1 ..  /\  /  \  /\/      \  /            \  /\/
.  /\/  \/    \/          \/              \/
.
Note that the j-th largest peak between two valleys at height 0 is also the partition number A000041(j).
Written as an irregular triangle in which row k has length 2*A138137(k), the sequence begins:
0,1;
0,1,2,1;
0,1,2,3,2,1;
0,1,2,1,2,3,4,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,4,3,2,1;
0,1,2,1,2,3,4,5,4,3,4,5,6,5,6,7,8,9,10,11,10,9,8,7,6,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,6,7,8,9,8,9,10,11,12,13,14,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1;
...

Links

Crossrefs

Column 1 is A000004. Both column 2 and the right border are in A000012. Both columns 3 and 5 are in A007395.
Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A229946.

A229946 Height of the peaks and the valleys in the Dyck path whose j-th ascending line segment has A141285(j) steps and whose j-th descending line segment has A194446(j) steps.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 2, 1, 5, 0, 3, 2, 7, 0, 2, 1, 5, 3, 6, 5, 11, 0, 3, 2, 7, 5, 9, 8, 15, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 0, 3, 2, 7, 5, 9, 8, 15, 11, 14, 13, 19, 17, 22, 21, 30, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 15, 19, 18, 25, 23, 29, 28, 33, 32, 42, 0
Offset: 0

Views

Author

Omar E. Pol, Nov 03 2013

Keywords

Comments

Also 0 together the alternating sums of A220517.
The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y]. See below:
.
. j Diagram 1 Partitions Diagram 2
. _ _ _ _
. 15 | _ | 7 _ |
. 14 | _ | | 4+3 _ | |
. 13 | _ | | 5+2 _ | |
. 12 | _| |_ | 3+2+2 _| |_ |
. 11 | _ | | 6+1 _ | |
. 10 | _| | | 3+3+1 _ | | |
. 9 | | | | 4+2+1 | | |
. 8 | |_ | | | 2+2+2+1 |_ | | |
. 7 | _ | | | 5+1+1 _ | | |
. 6 | _| | | | 3+2+1+1 _ | | | |
. 5 | | | | | 4+1+1+1 | | | |
. 4 | |_ | | | | 2+2+1+1+1 |_ | | | |
. 3 | | | | | | 3+1+1+1+1 | | | | |
. 2 | | | | | | | 2+1+1+1+1+1 | | | | | |
. 1 |||_|||_|_| 1+1+1+1+1+1+1 | | | | | | |
.
. 1 2 3 4 5 6 7
.
The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1.
Also the diagram has the property that it can be transformed in a Dyck path (see example).
The height of the peaks and the valleys of the infinite Dyck path give this sequence.
Q: Is this Dyck path a fractal?

Examples

			Illustration of initial terms (n = 0..21):
.                                                             11
.                                                             /
.                                                            /
.                                                           /
.                                   7                      /
.                                   /\                 6  /
.                     5            /  \           5    /\/
.                     /\          /    \          /\  / 5
.           3        /  \     3  /      \        /  \/
.      2    /\   2  /    \    /\/        \   2  /   3
.   1  /\  /  \  /\/      \  / 2          \  /\/
.   /\/  \/    \/ 1        \/              \/ 1
.  0 0   0     0           0               0
.
Note that the k-th largest peak between two valleys at height 0 is also A000041(k) and the next term is always 0.
.
Written as an irregular triangle in which row k has length 2*A187219(k), k >= 1, the sequence begins:
0,1;
0,2;
0,3;
0,2,1,5;
0,3,2,7;
0,2,1,5,3,6,5,11;
0,3,2,7,5,9,8,15;
0,2,1,5,3,6,5,11,7,12,11,15,14,22;
0,3,2,7,5,9,8,15,11,14,13,19,17,22,21,30;
0,2,1,5,3,6,5,11,7,12,11,15,14,22,15,19,18,25,23,29,28,33,32,42;
...

Links

Crossrefs

Column 1 is A000004. Right border gives A000041 for the positive integers.
Cf. A006128, A135010, A138137, A139582, A141285, A186412, A187219, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610.

Formula

a(0) = 0; a(n) = a(n-1) + (-1)^(n-1)*A220517(n), n >= 1.
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.