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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A106178 Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function, but omitting the trivial values of 1 at the margins of the table.

Original entry on oeis.org

2, 3, 1, 1, 1, 4, 5, 2, 9, 1, 6, 1, 1, 1, 2, 7, 1, 25, 1, 3, 1, 1, 1, 36, 1, 2, 1, 8, 1, 1, 49, 1, 5, 1, 27, 1, 10, 3, 1, 1, 6, 1, 1, 1, 2, 11, 1, 1, 2, 7, 1, 125, 4, 3, 1, 3, 1, 100, 1, 1, 1, 216, 1, 1, 1, 4, 13, 2, 121, 1, 3, 1, 343, 1, 5, 1, 9, 1, 14, 1, 9, 1, 10, 1, 1, 1, 6, 1, 2, 1, 2
Offset: 2

Views

Author

Jon Awbrey, May 28 2005

Keywords

Comments

This sequence is derived from A106177 by removing the "obvious" values of 1 at the margins of the triangular array.

Examples

			` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` n o m
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 . 1
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 . ` . 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 3 . ` . ` . 3
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 . ` . 2 . ` . 4
` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` 5 . ` . 3 . 1 . ` . 5
` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` 6 . ` . 1 . 1 . 4 . ` . 6
` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` 7 . ` . 5 . 2 . 9 . 1 . ` . 7
` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` 8 . ` . 6 . 1 . 1 . 1 . 2 . ` . 8
` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` 9 . ` . 7 . 1 . 25. 1 . 3 . 1 . ` . 9
` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` `10 . ` . 1 . 1 . 36. 1 . 2 . 1 . 8 . ` . 10
` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` `11 . ` . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 . ` . 11
` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` `12 . ` . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . ` . 12
` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` `13 . ` . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 . ` . 13
` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` `14 . ` . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 . ` . 14
` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` `15 . ` . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 . ` . 15
` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` `16 . ` . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 . ` . 16

Links

Crossrefs

Cf. A106177, A061396, A062504, A062537, A062860.

A111793 Triangle T(g, h) = number of rotes of weight g and height h, both in gammas.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 10, 8, 1, 24, 32, 16
Offset: 1

Views

Author

Jon Awbrey, Aug 26 2005, revised Aug 28 2005

Keywords

Comments

T(g, h) = |{positive integers m : A062537(m) = g and A109301(m) = h}|.
Row sums = A061396. Column sums = A109300. See A111792 for details.
Main diagonal T(j, j) = 2^(j-1) for j > 0, T(0, 0) = 1.

Examples

			Table T(g, h), omitting zeros, starts out as follows:
g\h| 0 ` 1 ` 2 ` 3 ` 4 ` 5
---+-----------------------
`0 | 1
`1 | ` ` 1
`2 | ` ` ` ` 2
`3 | ` ` ` ` 2 ` 4
`4 | ` ` ` ` 2 `10 ` 8
`5 | ` ` ` ` 1 `24 `32 `16

Links

Crossrefs

Cf. A007097, A050924, A061396, A062504, A062537, A062860.
Cf. A109300, A109301, A111791, A111792, A111794, A111800.

A112868 Positive integers sorted by rote weight and primal code characteristic.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 7, 8, 16, 10, 12, 14, 18, 11, 13, 17, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536, 22, 26, 34, 36, 38, 46, 50, 54, 98, 106, 125, 162, 2401, 15, 21, 29, 31, 37, 41, 43, 59, 61, 67, 83, 97, 103, 121, 131, 169, 227, 241, 243, 289, 311, 34, 361
Offset: 1

Views

Author

Jon Awbrey, Oct 13 2005

Keywords

Comments

Positive integers m sorted by g(m) = A062537(m) and q(m) = A108352(m).

Examples

			Primal Functions, Primal Codes, Sort Parameters, Subtotals
==========================================================
Primal Function | ` ` ` Primal Code ` = ` a | g q | s | t
==========================================================
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 1 | 0 1 | 1 | 1
==========================================================
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 2 | 1 0 | 1 | 1
==========================================================
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 3 | 2 2 | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 4 | 2 2 | 2 | 2
==========================================================
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 6 | 3 0 | ` |
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 9 | 3 0 | 2 |
----------------+---------------------------+-----+---+---
3:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 5 | 3 2 | ` |
4:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 7 | 3 2 | ` |
1:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 8 | 3 2 | ` |
1:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `16 | 3 2 | 4 | 6
==========================================================
1:1 3:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `10 | 4 0 | ` |
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `12 | 4 0 | ` |
1:1 4:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `14 | 4 0 | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `18 | 4 0 | 4 |
----------------+---------------------------+-----+---+---
5:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `11 | 4 2 | ` |
6:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `13 | 4 2 | ` |
7:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `17 | 4 2 | ` |
8:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `19 | 4 2 | ` |
9:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `23 | 4 2 | ` |
3:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `25 | 4 2 | ` |
2:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `27 | 4 2 | ` |
1:5 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `32 | 4 2 | ` |
4:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `49 | 4 2 | ` |
16:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `53 | 4 2 | ` |
1:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `64 | 4 2 | ` |
2:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `81 | 4 2 | ` |
1:7 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 128 | 4 2 | ` |
1:8 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 256 | 4 2 | ` |
1:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 512 | 4 2 | ` |
1:16` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 65536 | 4 2 |16 |20
==========================================================
a = this sequence
g = rote weight in gammas = A062537
q = primal code character = A108352
s = count in (g, q) class = A112869
t = count in weight class = A061396

Links

Crossrefs

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A111801, A112869.

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.

A112869 Triangle T(g, q) = number of rotes of weight g and primal code characteristic q.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 16, 13, 56, 4
Offset: 1

Views

Author

Jon Awbrey, Oct 13 2005

Keywords

Comments

T(g, q) = |{positive integers m : A062537(m) = g and A108352(m) = q}|.
This is the column that is labeled "s" in the tabulation of A112868.
Row sums = A061396.

Examples

			Table T(g, q), omitting empty cells, begins as follows:
g\q| 0 ` 1 ` 2 ` 3 ` 4 ` 5
---+-----------------------
`0 | ` ` 1 ` ` ` ` ` ` ` `
`1 | 1 ` ` ` ` ` ` ` ` ` `
`2 | ` ` ` ` 2 ` ` ` ` ` `
`3 | 2 ` ` ` 4 ` ` ` ` ` `
`4 | 4 ` ` `16 ` ` ` ` ` `
`5 |13 ` ` `56 ` 4 ` ` ` `

Links

Crossrefs

Cf. A061396, A062504, A062537, A062860, A106177, A106178.
Cf. A108352, A108353, A108370 to A108374, A111800, A111801.
Cf. A112846, A112868.

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

A111797 Triangle T(g, w) = number of rotes of weight g and wayage w.

Original entry on oeis.org

1, 1, 2, 5, 1, 16, 4, 56, 17
Offset: 1

Views

Author

Jon Awbrey, Sep 01 2005

Keywords

Comments

T(g, w) = |{positive integers m : A062537(m) = g and A001221(m) = w}|.
Row sums = A061396. See A111796 for definitions and further details.

Examples

			Table T(g, w), omitting zeros, begins as follows:
g\w| 0 ` 1 ` 2 ` 3 ` 4 ` 5
---+-----------------------
`0 | 1
`1 | ` ` 1
`2 | ` ` 2
`3 | ` ` 5 ` 1
`4 | ` `16 ` 4
`5 | ` `56 `17

Links

Crossrefs

Cf. A001221, A061396, A062504, A062537, A062860.
Cf. A111796, A111798, A111800.

A111798 Positive integers sorted by rote height (A109301) and omega (A001221).

Original entry on oeis.org

1, 2, 3, 4, 9, 6, 12, 18, 36, 5, 7, 8, 13, 16, 23, 25, 27, 37, 49, 61, 64, 81, 125, 151, 169, 343, 512, 529, 625, 729, 1369, 2197, 2401, 3721, 4096, 12167, 15625, 19683, 22801, 28561, 50653, 117649, 226981, 262144, 279841, 531441, 1874161, 1953125, 3442951
Offset: 1

Views

Author

Jon Awbrey, Sep 01 2005 - Sep 10 2005

Keywords

Comments

Positive integers m sorted by h(m) = A109301(m) and w(m) = A001221(m).
Defining the "wayage" of a rooted tree to be its root degree, the rote corresponding to the positive integer m has a wayage of w(m) = omega(m) = A001221(m).

Examples

			Table of Primal Functions, Codes, Sort Parameters and Subtotals
Primal Function | ` ` ` ` ` Primal Code ` ` = ` ` a | h w | s | t
----------------+-----------------------------------+-----+---+---
{ } ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 | 0 0 | 1 | 1
----------------+-----------------------------------+-----+---+---
1:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 | 1 1 | 1 | 1
----------------+-----------------------------------+-----+---+---
2:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 3 | 2 1 | ` |
1:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 | 2 1 | ` |
2:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 9 | 2 1 | 3 |
----------------+-----------------------------------+-----+---+---
1:1 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 6 | 2 2 | ` |
1:2 2:1 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `12 | 2 2 | ` |
1:1 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `18 | 2 2 | ` |
1:2 2:2 ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `36 | 2 2 | 4 | 7
----------------+-----------------------------------+-----+---+---
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
1:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 8 | 3 1 | ` |
1:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `16 | 3 1 | ` |
1:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `64 | 3 1 | ` |
1:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 512 | 3 1 | ` |
1:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `4096 | 3 1 | ` |
1:18` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `262144 | 3 1 | ` |
1:36` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 68719476736 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
2:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `27 | 3 1 | ` |
2:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `81 | 3 1 | ` |
2:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 729 | 3 1 | ` |
2:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 19683 | 3 1 | ` |
2:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `531441 | 3 1 | ` |
2:18` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 387420489 | 3 1 | ` |
2:36` ` ` ` ` ` | ` ` ` ` ` ` ` `150094635296999121 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
3:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 5 | 3 1 | ` |
4:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 7 | 3 1 | ` |
6:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `13 | 3 1 | ` |
9:1 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `23 | 3 1 | ` |
12:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `37 | 3 1 | ` |
18:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `61 | 3 1 | ` |
36:1` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 151 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
3:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `25 | 3 1 | ` |
4:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `49 | 3 1 | ` |
6:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 169 | 3 1 | ` |
9:2 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 529 | 3 1 | ` |
12:2` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `1369 | 3 1 | ` |
18:2` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `3721 | 3 1 | ` |
36:2` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 22801 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
3:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 125 | 3 1 | ` |
3:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 625 | 3 1 | ` |
3:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 15625 | 3 1 | ` |
3:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 1953125 | 3 1 | ` |
3:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 244140625 | 3 1 | ` |
3:18` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 3814697265625 | 3 1 | ` |
3:36` ` ` ` ` ` | ` ` ` `14551915228366851806640625 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
4:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 343 | 3 1 | ` |
4:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `2401 | 3 1 | ` |
4:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `117649 | 3 1 | ` |
4:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `40353607 | 3 1 | ` |
4:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 13841287201 | 3 1 | ` |
4:18` ` ` ` ` ` | ` ` ` ` ` ` ` ` `1628413597910449 | 3 1 | ` |
4:36` ` ` ` ` ` | ` 2651730845859653471779023381601 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
6:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `2197 | 3 1 | ` |
6:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 28561 | 3 1 | ` |
6:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 4826809 | 3 1 | ` |
6:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 10604499373 | 3 1 | ` |
6:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `23298085122481 | 3 1 | ` |
6:18` ` ` ` ` ` | ` ` ` ` ` ` 112455406951957393129 | 3 1 | ` |
6:36` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 13^36 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
9:3 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 12167 | 3 1 | ` |
9:4 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `279841 | 3 1 | ` |
9:6 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 148035889 | 3 1 | ` |
9:9 ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` 1801152661463 | 3 1 | ` |
9:12` ` ` ` ` ` | ` ` ` ` ` ` ` ` 21914624432020321 | 3 1 | ` |
9:18` ` ` ` ` ` | ` ` ` ` 3244150909895248285300369 | 3 1 | ` |
9:36` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 23^36 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
12:3` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 50653 | 3 1 | ` |
12:4` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 1874161 | 3 1 | ` |
12:6` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `2565726409 | 3 1 | ` |
12:9` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` 129961739795077 | 3 1 | ` |
12:12 ` ` ` ` ` | ` ` ` ` ` ` ` 6582952005840035281 | 3 1 | ` |
12:18 ` ` ` ` ` | ` ` 16890053810563300749953435929 | 3 1 | ` |
12:36 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 37^36 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
18:3` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `226981 | 3 1 | ` |
18:4` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `13845841 | 3 1 | ` |
18:6` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` 51520374361 | 3 1 | ` |
18:9` ` ` ` ` ` | ` ` ` ` ` ` ` ` 11694146092834141 | 3 1 | ` |
18:12 ` ` ` ` ` | ` ` ` ` ` `2654348974297586158321 | 3 1 | ` |
18:18 ` ` ` ` ` | 136753052840548005895349735207881 | 3 1 | ` |
18:36 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` 61^36 | 3 1 | ` |
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` |
36:3` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` 3442951 | 3 1 | ` |
36:4` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` 519885601 | 3 1 | ` |
36:6` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `11853911588401 | 3 1 | ` |
36:9` ` ` ` ` ` | ` ` ` ` ` ` `40812436757196811351 | 3 1 | ` |
36:12 ` ` ` ` ` | ` ` ` 140515219945627518837736801 | 3 1 | ` |
36:18 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `151^18 | 3 1 | ` |
36:36 ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` `151^36 | 3 1 |77 |
----------------+-----------------------------------+-----+---+---
The last part is left unsorted to show the method of construction.
a (when sorted ) = this sequence
h = rote height in gammas = A109301
w = rote wayage in gammas = A001221
s = count in (h, w) class = A111799
t = count in height class = A109300

Links

Crossrefs

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