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.

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A112846 Number of riffs on n or fewer nodes. Number of rotes on 2n+1 or fewer nodes.

Original entry on oeis.org

1, 2, 4, 10, 30, 103, 384, 1508, 6126, 25513, 108278, 466523, 2034981, 8968746, 39875940, 138760603, 178636543, 3026583484, 16028356176, 75647274620, 350111055991, 1618175863400, 7495933933620, 34821723061950
Offset: 0

Views

Author

Jon Awbrey, Oct 04 2005, based on calculations by Vladeta Jovovic and David W. Wilson

Keywords

Comments

Partial sums of A061396.

Links

Crossrefs

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800.
Cf. A112095, A112096, A112480, A112481.

A112095 Positive integers sorted by rote weight, rote height and rote wayage.

Original entry on oeis.org

1, 2, 3, 4, 9, 6, 5, 7, 8, 16, 12, 18, 13, 23, 25, 27, 49, 64, 81, 512, 10, 14, 11, 17, 19, 32, 53, 128, 256, 65536, 36, 37, 61, 125, 169, 343, 529, 625, 729, 2401, 4096, 19683, 262144, 15, 20, 21, 24, 26, 28, 46, 48, 50, 54, 98, 162, 29, 41, 43, 83, 97, 103, 121, 227
Offset: 1

Views

Author

Jon Awbrey, Sep 08 2005, corrected Oct 11 2005

Keywords

Comments

For positive integer m, the rote weight in gammas is g(m) = A062537(m), the rote height in gammas is h(m) = A109301(m) and the rote wayage or root degree is w(m) = omega(m) = A001221(m).

Examples

			Table of Primal Functions, Codes, Sort Parameters and Subtotals
================================================================
Primal Function | ` ` ` Primal Code ` = ` a | g h w | r | s | t
================================================================
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 1 | 0 0 0 | 1 | 1 | 1
================================================================
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 2 | 1 1 1 | 1 | 1 | 1
================================================================
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 3 | 2 2 1 | ` | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 4 | 2 2 1 | 2 | 2 | 2
================================================================
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 9 | 3 2 1 | 1 | ` |
----------------+---------------------------+-------+---+---+---
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 6 | 3 2 2 | 1 | 2 |
----------------+---------------------------+-------+---+---+---
3:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 5 | 3 3 1 | ` | ` |
4:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 7 | 3 3 1 | ` | ` |
1:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 8 | 3 3 1 | ` | ` |
1:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `16 | 3 3 1 | 4 | 4 | 6
================================================================
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `12 | 4 2 2 | ` | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `18 | 4 2 2 | 2 | 2 |
----------------+---------------------------+-------+---+---+---
6:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `13 | 4 3 1 | ` | ` |
9:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `23 | 4 3 1 | ` | ` |
3:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `25 | 4 3 1 | ` | ` |
2:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `27 | 4 3 1 | ` | ` |
4:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `49 | 4 3 1 | ` | ` |
1:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `64 | 4 3 1 | ` | ` |
2:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `81 | 4 3 1 | ` | ` |
1:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 512 | 4 3 1 | 8 | ` |
----------------+---------------------------+-------+---+---+---
1:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `10 | 4 3 2 | ` | ` |
1:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `14 | 4 3 2 | 2 |10 |
----------------+---------------------------+-------+---+---+---
5:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `11 | 4 4 1 | ` | ` |
7:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `17 | 4 4 1 | ` | ` |
8:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `19 | 4 4 1 | ` | ` |
1:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `32 | 4 4 1 | ` | ` |
16:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `53 | 4 4 1 | ` | ` |
1:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 128 | 4 4 1 | ` | ` |
1:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 256 | 4 4 1 | ` | ` |
1:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 65536 | 4 4 1 | 8 | 8 |20
================================================================
1:2 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `36 | 5 2 2 | 1 | 1 |
----------------+---------------------------+-------+---+---+---
12:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `37 | 5 3 1 | ` | ` |
18:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `61 | 5 3 1 | ` | ` |
3:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 125 | 5 3 1 | ` | ` |
6:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 169 | 5 3 1 | ` | ` |
4:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 343 | 5 3 1 | ` | ` |
9:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 529 | 5 3 1 | ` | ` |
3:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 625 | 5 3 1 | ` | ` |
2:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 729 | 5 3 1 | ` | ` |
4:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2401 | 5 3 1 | ` | ` |
1:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `4096 | 5 3 1 | ` | ` |
2:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 19683 | 5 3 1 | ` | ` |
1:18` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `262144 | 5 3 1 |12 | ` |
----------------+---------------------------+-------+---+---+---
2:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `15 | 5 3 2 | ` | ` |
1:2 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `20 | 5 3 2 | ` | ` |
2:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `21 | 5 3 2 | ` | ` |
1:3 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `24 | 5 3 2 | ` | ` |
1:1 6:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `26 | 5 3 2 | ` | ` |
1:2 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `28 | 5 3 2 | ` | ` |
1:1 9:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `46 | 5 3 2 | ` | ` |
1:4 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `48 | 5 3 2 | ` | ` |
1:1 3:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `50 | 5 3 2 | ` | ` |
1:1 2:3 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `54 | 5 3 2 | ` | ` |
1:1 4:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `98 | 5 3 2 | ` | ` |
1:1 2:4 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 162 | 5 3 2 |12 |24 |
----------------+---------------------------+-------+---+---+---
10:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `29 | 5 4 1 | ` | ` |
13:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `41 | 5 4 1 | ` | ` |
14:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `43 | 5 4 1 | ` | ` |
23:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `83 | 5 4 1 | ` | ` |
25:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `97 | 5 4 1 | ` | ` |
27:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 103 | 5 4 1 | ` | ` |
5:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 121 | 5 4 1 | ` | ` |
49:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 227 | 5 4 1 | ` | ` |
2:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 243 | 5 4 1 | ` | ` |
7:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 289 | 5 4 1 | ` | ` |
64:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 311 | 5 4 1 | ` | ` |
8:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 361 | 5 4 1 | ` | ` |
81:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 419 | 5 4 1 | ` | ` |
1:10` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `1024 | 5 4 1 | ` | ` |
2:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2187 | 5 4 1 | ` | ` |
16:2` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2809 | 5 4 1 | ` | ` |
512:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `3671 | 5 4 1 | ` | ` |
2:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `6561 | 5 4 1 | ` | ` |
1:13` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `8192 | 5 4 1 | ` | ` |
1:14` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 16384 | 5 4 1 | ` | ` |
1:23` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` 8388608 | 5 4 1 | ` | ` |
1:25` ` ` ` ` ` | ` ` ` ` ` ` ` ` `33554432 | 5 4 1 | ` | ` |
2:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` `43046721 | 5 4 1 | ` | ` |
1:27` ` ` ` ` ` | ` ` ` ` ` ` ` ` 134217728 | 5 4 1 | ` | ` |
1:49` ` ` ` ` ` | ` ` ` ` ` 562949953421312 | 5 4 1 | ` | ` |
1:64` ` ` ` ` ` | ` ` `18446744073709551616 | 5 4 1 | ` | ` |
1:81` ` ` ` ` ` | 2417851639229258349412352 | 5 4 1 | ` | ` |
1:512 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^512 | 5 4 1 |28 | ` |
----------------+---------------------------+-------+---+---+---
1:1 5:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `22 | 5 4 2 | ` | ` |
1:1 7:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `34 | 5 4 2 | ` | ` |
1:1 8:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `38 | 5 4 2 | ` | ` |
1:1 16:1` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 106 | 5 4 2 | 4 |32 |
----------------+---------------------------+-------+---+---+---
11:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `31 | 5 5 1 | ` | ` |
17:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `59 | 5 5 1 | ` | ` |
19:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `67 | 5 5 1 | ` | ` |
32:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 131 | 5 5 1 | ` | ` |
53:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 241 | 5 5 1 | ` | ` |
128:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 719 | 5 5 1 | ` | ` |
256:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `1619 | 5 5 1 | ` | ` |
1:11` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2048 | 5 5 1 | ` | ` |
1:17` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `131072 | 5 5 1 | ` | ` |
1:19` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `524288 | 5 5 1 | ` | ` |
65536:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` `821641 | 5 5 1 | ` | ` |
1:32` ` ` ` ` ` | ` ` ` ` ` ` ` `4294967296 | 5 5 1 | ` | ` |
1:53` ` ` ` ` ` | ` ` ` ` `9007199254740992 | 5 5 1 | ` | ` |
1:128 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^128 | 5 5 1 | ` | ` |
1:256 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^256 | 5 5 1 | ` | ` |
1:65536 ` ` ` ` | ` ` ` ` ` ` ` ` ` 2^65536 | 5 5 1 |16 |16 |73
================================================================
a = this sequence
g = rote weight in gammas = A062537
h = rote height in gammas = A109301
w = rote wayage in gammas = A001221
r = number in (g,h,w) set = A112096
s = count in (g, h) class = A111793
t = count in weight class = A061396

Links

Crossrefs

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800, A112096.

A112096 Tetrahedron T(g, h, w) = number of rotes of weight g, height h, wayage w.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 2, 8, 2, 8, 1, 12, 12, 28, 4, 16
Offset: 1

Views

Author

Jon Awbrey, Sep 08 2005, revised Sep 27 2005

Keywords

Comments

T(g, h, w) = |{m : A062537(m) = g, A109301(m) = h, A001221(m) = w}|.
This is the column that is labeled "r" in the tabulation of A112095.
g = h > 0 implies w = 1 and T(j, j, 1) = 2^(j-1) = A000079(j-1).

Examples

			Table T(g, h, w), omitting empty cells, starts out as follows:
g\(h,w) | (0,0) (1,1) (2,1) (2,2) (3,1) (3,2) (4,1) (4,2) (5,1)
--------+-------------------------------------------------------
0 ` ` ` | ` 1
1 ` ` ` | ` ` ` ` 1
2 ` ` ` | ` ` ` ` ` ` ` 2
3 ` ` ` | ` ` ` ` ` ` ` 1 ` ` 1 ` ` 4
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` 2 ` ` 8 ` ` 2 ` ` 8
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` 1 ` `12 ` `12 ` `28 ` ` 4 ` `16

Links

Crossrefs

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800, A112095.

A112871 Triangle T(h, q) = number of rotes of height h and quench q.

Original entry on oeis.org

1, 1, 5, 2
Offset: 1

Views

Author

Jon Awbrey, Oct 14 2005

Keywords

Comments

T(h, q) = |{positive integers m : A109301(m) = h and A108352(m) = q}|.
This is the column that is labeled "s" in the tabulation of A112870.
q(m) = quench(m) = A108352(m) = primal code characteristic of m.

Examples

			Table T(h, q), omitting empty cells, begins as follows:
h\q| 0 ` 1 ` 2
---+----------
`0 | ` ` 1 ` `
`1 | 1 ` ` ` `
`2 | 5 ` ` ` 2
Row sums = A109300.

Links

Crossrefs

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A109300, A109301.
Cf. A111791 to A111801, A112846, A112868, A112869, A112870.

Extensions

Too short to be interesting - hope more terms can be supplied soon! - N. J. A. Sloane

A113197 Positive integers sorted by rote weight, rote height and rote quench.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 7, 8, 16, 12, 18, 10, 14, 13, 23, 25, 27, 49, 64, 81, 512, 11, 17, 19, 32, 53, 128, 256, 65536, 36, 26, 46, 50, 54, 98, 125, 162, 2401, 15, 21, 37, 61, 169, 343, 529, 625, 729, 4096, 19683, 262144, 20, 24, 28, 48, 22, 34, 38, 106, 29, 41, 43, 83, 97
Offset: 1

Views

Author

Jon Awbrey, Oct 18 2005

Keywords

Comments

For positive integer m, the rote weight in gammas is g(m) = A062537(m), the rote height in gammas is h(m) = A109301(m) and the rote quench or primal code characteristic is q(m) = A108352(m).

Examples

			Primal Functions, Primal Codes, Sort Parameters and Subtotals
================================================================
Primal Function | ` ` ` Primal Code ` = ` a | g h q | r | s | t
================================================================
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 1 | 0 0 1 | 1 | 1 | 1
================================================================
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 2 | 1 1 0 | 1 | 1 | 1
================================================================
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 3 | 2 2 2 | ` | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 4 | 2 2 2 | 2 | 2 | 2
================================================================
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 6 | 3 2 0 | ` | ` |
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 9 | 3 2 0 | 2 | 2 |
----------------+---------------------------+-------+---+---+---
3:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 5 | 3 3 2 | ` | ` |
4:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 7 | 3 3 2 | ` | ` |
1:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 8 | 3 3 2 | ` | ` |
1:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `16 | 3 3 2 | 4 | 4 | 6
================================================================
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `12 | 4 2 0 | ` | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `18 | 4 2 0 | 2 | 2 |
----------------+---------------------------+-------+---+---+---
1:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `10 | 4 3 0 | ` | ` |
1:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `14 | 4 3 0 | 2 | ` |
----------------+---------------------------+-------+---+---+---
6:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `13 | 4 3 2 | ` | ` |
9:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `23 | 4 3 2 | ` | ` |
3:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `25 | 4 3 2 | ` | ` |
2:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `27 | 4 3 2 | ` | ` |
4:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `49 | 4 3 2 | ` | ` |
1:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `64 | 4 3 2 | ` | ` |
2:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `81 | 4 3 2 | ` | ` |
1:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 512 | 4 3 2 | 8 |10 |
----------------+---------------------------+-------+---+---+---
5:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `11 | 4 4 2 | ` | ` |
7:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `17 | 4 4 2 | ` | ` |
8:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `19 | 4 4 2 | ` | ` |
1:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `32 | 4 4 2 | ` | ` |
16:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `53 | 4 4 2 | ` | ` |
1:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 128 | 4 4 2 | ` | ` |
1:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 256 | 4 4 2 | ` | ` |
1:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 65536 | 4 4 2 | 8 | 8 |20
================================================================
a = this sequence
g = rote weight in gammas = A062537
h = rote height in gammas = A109301
q = primal code character = A108352
r = number in (g,h,q) set = A113198
s = count in (g, h) class = A111793
t = count in weight class = A061396

Links

Crossrefs

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A109300, A109301.
Cf. A111791 to A111801, A112846, A112868 to A112871, A113198.

A113198 Tetrahedron T(g, h, q) = number of rotes of weight g, height h, quench q.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 2, 8, 8, 1, 8, 12, 4, 4, 28, 16
Offset: 1

Views

Author

Jon Awbrey, Oct 18 2005

Keywords

Comments

T(g, h, q) = |{m : A062537(m) = g, A109301(m) = h, A108352(m) = q}|.
This is the column that is labeled "r" in the tabulation of A113197.

Examples

			Table T(g, h, q), omitting empty cells, starts out as follows:
--------+------------------------------------------------------------
g\(h,q) | (0,1) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` | ` ` ` (1,0) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` ` (2,0) (2,2) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` (3,0) (3,2) (3,3) ` ` ` ` ` ` ` ` `
` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` (4,0) (4,2) ` ` `
` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` (5,2)
========+============================================================
0 ` ` ` | ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
1 ` ` ` | ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
2 ` ` ` | ` ` ` ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
3 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
3 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 ` ` ` ` ` ` ` ` ` ` ` ` `
--------+------------------------------------------------------------
4 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 2 ` ` 8 ` ` ` ` ` ` ` ` ` ` ` ` `
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 8 ` ` ` `
--------+------------------------------------------------------------
5 ` ` ` | ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 8 ` `12 ` ` 4 ` ` ` ` ` ` ` ` ` `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 ` `28 ` ` ` `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `16 `
--------+------------------------------------------------------------
Row sums = A111793. Horizontal section sums = A061396.

Links

Crossrefs

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A109300, A109301.
Cf. A111791 to A111801, A112846, A112868 to A112871, A113197.

A109299 Primal codes of canonical finite permutations on positive integers.

Original entry on oeis.org

1, 2, 12, 18, 360, 540, 600, 1350, 1500, 2250, 75600, 105840, 113400, 126000, 158760, 246960, 283500, 294000, 315000, 411600, 472500, 555660, 735000, 864360, 992250, 1296540, 1389150, 1440600, 1653750, 2572500, 3241350, 3601500, 3858750
Offset: 1

Views

Author

Jon Awbrey, Jul 09 2005

Keywords

Comments

A canonical finite permutation on positive integers is a bijective mapping of [n] = {1, ..., n} to itself, counting the empty mapping as a permutation of the empty set.
From Rémy Sigrist, Sep 18 2021: (Start)
As usual with lists, the terms of the sequence are given in ascending order.
Equivalently, these are the numbers m such that A001221(m) = A051903(m) = A061395(m) = A071625(m).
This sequence has connections with A175061; here the prime factorizations, there the run-lengths in binary expansions, encode finite permutations.
There are m! terms with m distinct prime factors, the least one being A006939(m) and the greatest one being A076954(m); these m! terms are not necessarily contiguous. (End)

Examples

			Writing (prime(i))^j as i:j, we have this table:
Primal Codes of Canonical Finite Permutations
        1 = { }
        2 = 1:1
       12 = 1:2 2:1
       18 = 1:1 2:2
      360 = 1:3 2:2 3:1
      540 = 1:2 2:3 3:1
      600 = 1:3 2:1 3:2
     1350 = 1:1 2:3 3:2
     1500 = 1:2 2:1 3:3
     2250 = 1:1 2:2 3:3
    75600 = 1:4 2:3 3:2 4:1
   105840 = 1:4 2:3 3:1 4:2
   113400 = 1:3 2:4 3:2 4:1
   126000 = 1:4 2:2 3:3 4:1
   158760 = 1:3 2:4 3:1 4:2
   246960 = 1:4 2:2 3:1 4:3
   283500 = 1:2 2:4 3:3 4:1
   294000 = 1:4 2:1 3:3 4:2
   315000 = 1:3 2:2 3:4 4:1
   411600 = 1:4 2:1 3:2 4:3
   472500 = 1:2 2:3 3:4 4:1
   555660 = 1:2 2:4 3:1 4:3
   735000 = 1:3 2:1 3:4 4:2
   864360 = 1:3 2:2 3:1 4:4
   992250 = 1:1 2:4 3:3 4:2
  1296540 = 1:2 2:3 3:1 4:4
  1389150 = 1:1 2:4 3:2 4:3
  1440600 = 1:3 2:1 3:2 4:4
  1653750 = 1:1 2:3 3:4 4:2
  2572500 = 1:2 2:1 3:4 4:3
  3241350 = 1:1 2:3 3:2 4:4
  3601500 = 1:2 2:1 3:3 4:4
  3858750 = 1:1 2:2 3:4 4:3
  5402250 = 1:1 2:2 3:3 4:4

References

  • Suggested by Franklin T. Adams-Watters

Links

Crossrefs

Cf. A001221, A006939, A051903, A061395, A071625, A076954, A106177, A108352, A108371, A109297, A109298, A109301, A175061, A347758.

Programs

  • PARI
    \\ See Links section.
  • PARI
    is(n) = { my (f=factor(n), p=f[,1]~, e=f[,2]~); Set(e)==[1..#e] && (#p==0 || p[#p]==prime(#p)) } \\ Rémy Sigrist, Sep 18 2021

Extensions

Offset changed to 1 and data corrected by Rémy Sigrist, Sep 18 2021

A111794 Integers whose rote weight and rote height are equal, sorted by the equated value.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 16, 11, 17, 19, 32, 53, 128, 256, 65536, 31, 59, 67, 131, 241, 719, 1619, 2048, 131072, 524288, 821641, 4294967296, 9007199254740992
Offset: 1

Views

Author

Jon Awbrey, Aug 28 2005

Keywords

Comments

The number of integers m whose rote weight, g(m) = A062537(m) and rote height, h(m) = A109301(m), are both equal to j is 2^(j-1) for j > 0 and 1 for j = 0, as enumerated by the main diagonal of the array shown with sequence A111793.

Examples

			Triangle whose j^th row lists the integers m with g(m) = h(m) = j
j | m such that g(m) = h(m) = j
--+-------------------------------------------------------
0 | 1
1 | 2
2 | 3 4
3 | 5 7 8 16
4 | 11 17 19 32 53 128 256 65536
5 | 31 59 67 131 241 719 1619 2048 131072 524288 821641
` | 4294967296 9007199254740992 2^128 2^256 2^65536

Links

Crossrefs

Cf. A007097, A050924, A061396, A062504, A062537, A062860.
Cf. A109300, A109301, A111791, A111792, A111793, A111800.

A112480 Positive integers sorted by rote weight, rote wagage and rote height.

Original entry on oeis.org

1, 2, 3, 4, 9, 5, 7, 8, 16, 6, 13, 23, 25, 27, 49, 64, 81, 512, 11, 17, 19, 32, 53, 128, 256, 65536, 12, 18, 10, 14, 37, 61, 125, 169, 343, 529, 625, 729, 2401, 4096, 19683, 262144, 29, 41, 43, 83, 97, 103, 121, 227, 243, 289, 311, 361, 419, 1024, 2187, 2809, 3671
Offset: 1

Views

Author

Jon Awbrey, Sep 27 2005

Keywords

Comments

For positive integer m, the rote weight in gammas is g(m) = A062537(m), the rote wayage or root degree is w(m) = omega(m) = A001221(m) and the rote height in gammas is h(m) = A109301(m).

Examples

			Table of Primal Functions, Codes, Sort Parameters and Subtotals
================================================================
Primal Function | ` ` ` Primal Code ` = ` a | g w h | r | s | t
================================================================
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 1 | 0 0 0 | 1 | 1 | 1
================================================================
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 2 | 1 1 1 | 1 | 1 | 1
================================================================
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 3 | 2 1 2 | ` | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 4 | 2 1 2 | 2 | 2 | 2
================================================================
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 9 | 3 1 2 | 1 | ` |
----------------+---------------------------+-------+---+---+---
3:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 5 | 3 1 3 | ` | ` |
4:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 7 | 3 1 3 | ` | ` |
1:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 8 | 3 1 3 | ` | ` |
1:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `16 | 3 1 3 | 4 | 5 |
----------------+---------------------------+-------+---+---+---
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 6 | 3 2 2 | 1 | 1 | 6
================================================================
6:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `13 | 4 1 3 | ` | ` |
9:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `23 | 4 1 3 | ` | ` |
3:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `25 | 4 1 3 | ` | ` |
2:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `27 | 4 1 3 | ` | ` |
4:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `49 | 4 1 3 | ` | ` |
1:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `64 | 4 1 3 | ` | ` |
2:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `81 | 4 1 3 | ` | ` |
1:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 512 | 4 1 3 | 8 | ` |
----------------+---------------------------+-------+---+---+---
5:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `11 | 4 1 4 | ` | ` |
7:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `17 | 4 1 4 | ` | ` |
8:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `19 | 4 1 4 | ` | ` |
1:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `32 | 4 1 4 | ` | ` |
16:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `53 | 4 1 4 | ` | ` |
1:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 128 | 4 1 4 | ` | ` |
1:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 256 | 4 1 4 | ` | ` |
1:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 65536 | 4 1 4 | 8 |16 |
----------------+---------------------------+-------+---+---+---
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `12 | 4 2 2 | ` | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `18 | 4 2 2 | 2 | ` |
----------------+---------------------------+-------+---+---+---
1:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `10 | 4 2 3 | ` | ` |
1:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `14 | 4 2 3 | 2 | 4 |20
================================================================
12:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `37 | 5 1 3 | ` | ` |
18:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `61 | 5 1 3 | ` | ` |
3:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 125 | 5 1 3 | ` | ` |
6:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 169 | 5 1 3 | ` | ` |
4:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 343 | 5 1 3 | ` | ` |
9:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 529 | 5 1 3 | ` | ` |
3:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 625 | 5 1 3 | ` | ` |
2:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 729 | 5 1 3 | ` | ` |
4:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2401 | 5 1 3 | ` | ` |
1:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `4096 | 5 1 3 | ` | ` |
2:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 19683 | 5 1 3 | ` | ` |
1:18` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `262144 | 5 1 3 |12 | ` |
----------------+---------------------------+-------+---+---+---
10:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `29 | 5 1 4 | ` | ` |
13:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `41 | 5 1 4 | ` | ` |
14:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `43 | 5 1 4 | ` | ` |
23:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `83 | 5 1 4 | ` | ` |
25:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `97 | 5 1 4 | ` | ` |
27:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 103 | 5 1 4 | ` | ` |
5:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 121 | 5 1 4 | ` | ` |
49:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 227 | 5 1 4 | ` | ` |
2:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 243 | 5 1 4 | ` | ` |
7:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 289 | 5 1 4 | ` | ` |
64:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 311 | 5 1 4 | ` | ` |
8:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 361 | 5 1 4 | ` | ` |
81:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 419 | 5 1 4 | ` | ` |
1:10` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `1024 | 5 1 4 | ` | ` |
2:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2187 | 5 1 4 | ` | ` |
16:2` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2809 | 5 1 4 | ` | ` |
512:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `3671 | 5 1 4 | ` | ` |
2:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `6561 | 5 1 4 | ` | ` |
1:13` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `8192 | 5 1 4 | ` | ` |
1:14` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 16384 | 5 1 4 | ` | ` |
1:23` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` 8388608 | 5 1 4 | ` | ` |
1:25` ` ` ` ` ` | ` ` ` ` ` ` ` ` `33554432 | 5 1 4 | ` | ` |
2:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` `43046721 | 5 1 4 | ` | ` |
1:27` ` ` ` ` ` | ` ` ` ` ` ` ` ` 134217728 | 5 1 4 | ` | ` |
1:49` ` ` ` ` ` | ` ` ` ` ` 562949953421312 | 5 1 4 | ` | ` |
1:64` ` ` ` ` ` | ` ` `18446744073709551616 | 5 1 4 | ` | ` |
1:81` ` ` ` ` ` | 2417851639229258349412352 | 5 1 4 | ` | ` |
1:512 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^512 | 5 1 4 |28 | ` |
----------------+---------------------------+-------+---+---+---
11:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `31 | 5 1 5 | ` | ` |
17:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `59 | 5 1 5 | ` | ` |
19:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `67 | 5 1 5 | ` | ` |
32:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 131 | 5 1 5 | ` | ` |
53:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 241 | 5 1 5 | ` | ` |
128:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 719 | 5 1 5 | ` | ` |
256:1 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `1619 | 5 1 5 | ` | ` |
1:11` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `2048 | 5 1 5 | ` | ` |
1:17` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `131072 | 5 1 5 | ` | ` |
1:19` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `524288 | 5 1 5 | ` | ` |
65536:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` `821641 | 5 1 5 | ` | ` |
1:32` ` ` ` ` ` | ` ` ` ` ` ` ` `4294967296 | 5 1 5 | ` | ` |
1:53` ` ` ` ` ` | ` ` ` ` `9007199254740992 | 5 1 5 | ` | ` |
1:128 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^128 | 5 1 5 | ` | ` |
1:256 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 2^256 | 5 1 5 | ` | ` |
1:65536 ` ` ` ` | ` ` ` ` ` ` ` ` ` 2^65536 | 5 1 5 |16 |56 |
----------------+---------------------------+-------+---+---+---
1:2 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `36 | 5 2 2 | 1 | ` |
----------------+---------------------------+-------+---+---+---
2:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `15 | 5 2 3 | ` | ` |
1:2 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `20 | 5 2 3 | ` | ` |
2:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `21 | 5 2 3 | ` | ` |
1:3 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `24 | 5 2 3 | ` | ` |
1:1 6:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `26 | 5 2 3 | ` | ` |
1:2 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `28 | 5 2 3 | ` | ` |
1:1 9:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `46 | 5 2 3 | ` | ` |
1:4 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `48 | 5 2 3 | ` | ` |
1:1 3:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `50 | 5 2 3 | ` | ` |
1:1 2:3 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `54 | 5 2 3 | ` | ` |
1:1 4:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `98 | 5 2 3 | ` | ` |
1:1 2:4 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 162 | 5 2 3 |12 | ` |
----------------+---------------------------+-------+---+---+---
1:1 5:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `22 | 5 2 4 | ` | ` |
1:1 7:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `34 | 5 2 4 | ` | ` |
1:1 8:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `38 | 5 2 4 | ` | ` |
1:1 16:1` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 106 | 5 2 4 | 4 |17 |73
================================================================
a = this sequence
g = rote weight in gammas = A062537
w = rote wayage in gammas = A001221
h = rote height in gammas = A109301
r = number in (g,h,w) set = A112481
s = count in (g, w) class = A111797
t = count in weight class = A061396

Links

Crossrefs

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800.
Cf. A112095, A112096, A112481.

A112481 Tetrahedron T(g, w, h) = number of rotes of weight g, wayage w, height h.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 8, 8, 2, 2, 12, 28, 16, 1, 12, 4
Offset: 1

Views

Author

Jon Awbrey, Sep 27 2005

Keywords

Comments

T(g, w, h) = |{m : A062537(m) = g, A001221(m) = w, A109301(m) = h}|.
This is the column that is labeled "r" in the tabulation of A112480.
a(n) is a permutation of the elements in A112096.
g = h > 0 implies w = 1 and T(j, 1, j) = 2^(j-1) = A000079(j-1).

Examples

			Table T(g, w, h), omitting empty cells, starts out as follows:
--------+-------------------------------------------------------
g\(w,h) | (0,0) (1,1) (1,2) ` ` ` (1,3) ` ` ` (1,4) ` ` ` (1,5)
` ` ` ` | ` ` ` ` ` ` ` ` ` (2,2) ` ` ` (2,3) ` ` ` (2,4) ` ` `
========+=======================================================
0 ` ` ` | ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
1 ` ` ` | ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
2 ` ` ` | ` ` ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
3 ` ` ` | ` ` ` ` ` ` ` 1 ` ` ` ` ` 4 ` ` ` ` ` ` ` ` ` ` ` ` `
3 ` ` ` | ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 8 ` ` ` ` ` 8 ` ` ` ` ` ` `
4 ` ` ` | ` ` ` ` ` ` ` ` ` ` 2 ` ` ` ` ` 2 ` ` ` ` ` ` ` ` ` `
--------+-------------------------------------------------------
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `12 ` ` ` ` `28 ` ` ` ` `16 `
5 ` ` ` | ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` `12 ` ` ` ` ` 4 ` ` ` `
--------+-------------------------------------------------------
Row sums = A111797. Horizontal section sums = A061396.

Links

Crossrefs

Cf. A000079, A001221, A007097, A014221, A050924.
Cf. A061396, A062504, A062537, A062860, A106177.
Cf. A109300, A109301, A111791 to A111800.
Cf. A112095, A112096, A112480.
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