User:Jon Awbrey/TABLES

A061396

Plain Wiki Table

Large Scale

${\text{Table 1.}}~~{\text{Prime Factorizations, Riffs, and Rotes}}$

${\text{Integer}}\!$	${\text{Factorization}}\!$	${\text{Notation}}\!$	${\text{Riff Digraph}}\!$	${\text{Rote Graph}}\!$	${\text{Traversal}}\!$
$1\!$	$1\!$
$2\!$	${\text{p}}_{1}^{1}\!$	${\text{p}}\!$			$((~))$
$3\!$	${\begin{array}{lll}{\text{p}}_{2}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}_{\text{p}}\!$	$\cdots$		$(((~))(~))$
$4\!$	${\begin{array}{lll}{\text{p}}_{1}^{2}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}^{\text{p}}\!$	$\cdots$		$((((~))))$
$5\!$	${\begin{array}{lll}{\text{p}}_{3}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[6pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}_{\text{p}}}\!$	$\cdots$		$((((~))(~))(~))$
$6\!$	${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}{\text{p}}_{\text{p}}\!$	$\cdots$		$((~))(((~))(~))$
$7\!$	${\begin{array}{lll}{\text{p}}_{4}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[6pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}^{\text{p}}}\!$	$\cdots$		$(((((~))))(~))$
$8\!$	${\begin{array}{lll}{\text{p}}_{1}^{3}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{1}}\\[6pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}$	${\text{p}}^{{\text{p}}_{\text{p}}}\!$	$\cdots$		$(((((~))(~))))$
$9\!$	${\begin{array}{lll}{\text{p}}_{2}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}_{\text{p}}^{\text{p}}\!$	$\cdots$		$(((~))(((~))))$
$16\!$	${\begin{array}{lll}{\text{p}}_{1}^{4}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{2}}\\[6pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}$	${\text{p}}^{{\text{p}}^{\text{p}}}\!$	$\cdots$		$((((((~))))))$

Small Scale

${\text{Table 1.}}~~{\text{Prime Factorizations, Riffs, and Rotes}}$

${\text{Integer}}\!$	${\text{Factorization}}\!$	${\text{Notation}}\!$	${\text{Riff Digraph}}\!$	${\text{Rote Graph}}\!$	${\text{Traversal}}\!$
$1\!$	$1\!$
$2\!$	${\text{p}}_{1}^{1}\!$	${\text{p}}\!$			$((~))$
$3\!$	${\begin{array}{lll}{\text{p}}_{2}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}_{\text{p}}\!$	$\cdots$		$(((~))(~))$
$4\!$	${\begin{array}{lll}{\text{p}}_{1}^{2}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}^{\text{p}}\!$	$\cdots$		$((((~))))$
$5\!$	${\begin{array}{lll}{\text{p}}_{3}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[6pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}_{\text{p}}}\!$	$\cdots$		$((((~))(~))(~))$
$6\!$	${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}{\text{p}}_{\text{p}}\!$	$\cdots$		$((~))(((~))(~))$
$7\!$	${\begin{array}{lll}{\text{p}}_{4}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[6pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}^{\text{p}}}\!$	$\cdots$		$(((((~))))(~))$
$8\!$	${\begin{array}{lll}{\text{p}}_{1}^{3}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{1}}\\[6pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}$	${\text{p}}^{{\text{p}}_{\text{p}}}\!$	$\cdots$		$(((((~))(~))))$
$9\!$	${\begin{array}{lll}{\text{p}}_{2}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}_{\text{p}}^{\text{p}}\!$	$\cdots$		$(((~))(((~))))$
$16\!$	${\begin{array}{lll}{\text{p}}_{1}^{4}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{2}}\\[6pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}$	${\text{p}}^{{\text{p}}^{\text{p}}}\!$	$\cdots$		$((((((~))))))$

Nested Wiki Table

Large Scale

${\text{Table 1.}}~~{\text{Prime Factorizations, Riffs, and Rotes}}$

${\text{Integer}}\!$ ${\text{Factorization}}\!$ ${\text{Notation}}\!$ ${\text{Riff Digraph}}\!$ ${\text{Rote Graph}}\!$ ${\text{Traversal}}\!$

$1\!$ $1\!$

$2\!$ ${\text{p}}_{1}^{1}\!$ ${\text{p}}\!$ $((~))$

$3\!$ ${\begin{array}{lll}{\text{p}}_{2}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$ ${\text{p}}_{\text{p}}\!$ $\cdots$ $(((~))(~))$ $4\!$ ${\begin{array}{lll}{\text{p}}_{1}^{2}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}\end{array}}$ ${\text{p}}^{\text{p}}\!$ $\cdots$ $((((~))))$

$5\!$ ${\begin{array}{lll}{\text{p}}_{3}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[10pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}$ ${\text{p}}_{{\text{p}}_{\text{p}}}\!$ $\cdots$ $((((~))(~))(~))$ $6\!$ ${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$ ${\text{p}}{\text{p}}_{\text{p}}\!$ $\cdots$ $((~))(((~))(~))$ $7\!$ ${\begin{array}{lll}{\text{p}}_{4}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[10pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}$ ${\text{p}}_{{\text{p}}^{\text{p}}}\!$ $\cdots$ $(((((~))))(~))$ $8\!$ ${\begin{array}{lll}{\text{p}}_{1}^{3}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{1}}\\[10pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}$ ${\text{p}}^{{\text{p}}_{\text{p}}}\!$ $\cdots$ $(((((~))(~))))$ $9\!$ ${\begin{array}{lll}{\text{p}}_{2}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}$ ${\text{p}}_{\text{p}}^{\text{p}}\!$ $\cdots$ $(((~))(((~))))$ $16\!$ ${\begin{array}{lll}{\text{p}}_{1}^{4}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{2}}\\[10pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}$ ${\text{p}}^{{\text{p}}^{\text{p}}}\!$ $\cdots$ $((((((~))))))$

Small Scale

${\text{Table 1.}}~~{\text{Prime Factorizations, Riffs, and Rotes}}$

${\text{Integer}}\!$ ${\text{Factorization}}\!$ ${\text{Notation}}\!$ ${\text{Riff Digraph}}\!$ ${\text{Rote Graph}}\!$ ${\text{Traversal}}\!$

$1\!$ $1\!$

$2\!$ ${\text{p}}_{1}^{1}\!$ ${\text{p}}\!$ $((~))$

$3\!$ ${\begin{array}{lll}{\text{p}}_{2}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$ ${\text{p}}_{\text{p}}\!$ $\cdots$ $(((~))(~))$ $4\!$ ${\begin{array}{lll}{\text{p}}_{1}^{2}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}\end{array}}$ ${\text{p}}^{\text{p}}\!$ $\cdots$ $((((~))))$

$5\!$ ${\begin{array}{lll}{\text{p}}_{3}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[10pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}$ ${\text{p}}_{{\text{p}}_{\text{p}}}\!$ $\cdots$ $((((~))(~))(~))$ $6\!$ ${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$ ${\text{p}}{\text{p}}_{\text{p}}\!$ $\cdots$ $((~))(((~))(~))$ $7\!$ ${\begin{array}{lll}{\text{p}}_{4}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[10pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}$ ${\text{p}}_{{\text{p}}^{\text{p}}}\!$ $\cdots$ $(((((~))))(~))$ $8\!$ ${\begin{array}{lll}{\text{p}}_{1}^{3}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{1}}\\[10pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}$ ${\text{p}}^{{\text{p}}_{\text{p}}}\!$ $\cdots$ $(((((~))(~))))$ $9\!$ ${\begin{array}{lll}{\text{p}}_{2}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}$ ${\text{p}}_{\text{p}}^{\text{p}}\!$ $\cdots$ $(((~))(((~))))$ $16\!$ ${\begin{array}{lll}{\text{p}}_{1}^{4}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{2}}\\[10pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}$ ${\text{p}}^{{\text{p}}^{\text{p}}}\!$ $\cdots$ $((((((~))))))$

Old ASCII Version

Illustration of initial terms of A061396
Jon Awbrey (jawbrey(AT)oakland.edu)

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_(p_p)   @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p_(p^p)   @     o      @            ((((())))())
|                                        ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =                    /   \       |
|           p_1^p_(p_1)       p^p_p     @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =                  /           |
|           p_(p_1)^(p_1)     p_p^p     @            @            ((())((())))
|                                        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =                       ^          o---o
|           p_1^(p_1^2) =                /           |
|           p_1^(p_1^p_1)     p^(p^p)   @            @            (((((())))))
|
o--------------------------------------------------------------------------------

Further Comments:

Here are a couple more pages from my notes,
where it looks like I first arrived at the
generating function, and also carried out
some brute force enumerations of riffs.

I am going to experiment with a different way of
transcribing indices and powers into a plaintext.

|                jj
|              p<
|      j      /  ji
|    p<     p<         etc.
|      i      \  ij
|              p<
|                ii

-------------------------------------------------------

1978-11-06

Generating Function

| R(x) = 1 + x + 2x^2 + ...
|
|      =   1 + x.x^0 (1 + x + 2x^2 + ...)
|        . 1 + x.x^1 (1 + x + 2x^2 + ...)
|        . 1 + x.x^2 (1 + x + 2x^2 + ...)
|        . 1 + x.x^2 (1 + x + 2x^2 + ...)
|        . ...
|
|      = 1 + x + 2x^2 + ...
|
| Product over (i = 0 to infinity) of (1 + x.x^i.R(x))^R_i  =  R(x)

-------------------------------------------------------

1978-11-10

Brute force enumeration of R_n

| 4 p's
|
|       p
|     p<        p_p                 p                    p
|   p<        p<        p p_p     p<_p     p_p_p     p_p<
| p<        p<        p<        p<       p<        p<
|
|
|       p
|     p<        p_p                 p                    p
| p_p<      p_p<      p<        p_p<_p   p_p_p_p   p_p_p<
|                       p p_p
|
|
|     p
|   p<        p_p       p         p        p           p
| p<        p<        p<        p<       p<  p<    p p<
|   p         p         p_p       p^p          p       p
|
|
| p p_p_p   p p<
|               p^p
|

Altogether, 20 riffs of weight 4.

| o---------------------o---------------------o---------------------o
| | 3                   | 4                   | 5                   |
| o---------------------o---------------------o---------------------|
| | // // 2             | 10, 3, 1, 6         | 36, 10, 2, 3, 2, 20 |
| o---------------------o---------------------o---------------------|
| |                     | 0^1 4^1,            |                     |
| |                     | 1^1 3^1,            |                     |
| |                     | 2^2,                |                     |
| |                     | 4^1 0^1             |                     |
| o---------------------o---------------------o---------------------o
| | 6                   | 20                  | 73                  |
| o---------------------o---------------------o---------------------o
|

-------------------------------------------------------

Here are the number values of the riffs on 4 nodes:

o----------------------------------------------------------------------
|
|       p
|     p<        p_p                 p                    p
|   p<        p<        p p_p     p<_p     p_p_p     p_p<
| p<        p<        p<        p<       p<        p<
|
| 2^16      2^8       2^6       2^9      2^5       2^7
| 65536     256       64        512      32        128
o----------------------------------------------------------------------
|
|       p
|     p<        p_p                 p                    p
| p_p<      p_p<      p<        p_p<_p   p_p_p_p   p_p_p<
|                       p p_p
|
| p_16      p_8       p_6       p_9      p_5       p_7
| 53        19        13        23       11        17
o----------------------------------------------------------------------
|
|     p
|   p<        p_p       p         p                    p
| p<        p<        p<        p<       p^p p_p   p p<
|   p         p         p_p       p^p                  p
|
| 3^4       3^3       5^2       7^2
| 81        27        25        49       12        18
o----------------------------------------------------------------------
|
| p p_p_p   p p<
|               p^p
|
| 10        14 
o----------------------------------------------------------------------

For ease of reference, I include the previous table
of smaller riffs and rotes, redone in the new style.

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_p_p     @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p<        @     o      @            ((((())))())
|                               p^p      ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =           p_p      /   \       |
|           p_1^p_(p_1)       p<        @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =         p        /           |
|           p_(p_1)^(p_1)     p<        @            @            ((())((())))
|                               p        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =               p       ^          o---o
|           p_1^(p_1^2) =       p<       /           |
|           p_1^(p_1^p_1)     p<        @            @            (((((())))))
|
o--------------------------------------------------------------------------------

(later)

Expanded version of first table:

o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_p_p     @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p<        @     o      @            ((((())))())
|                               p^p      ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =           p_p      /   \       |
|           p_1^p_(p_1)       p<        @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =         p        /           |
|           p_(p_1)^(p_1)     p<        @            @            ((())((())))
|                               p        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =               p       ^          o---o
|           p_1^(p_1^2) =       p<       /           |
|           p_1^(p_1^p_1)     p<        @            @            (((((())))))
|
o--------------------------------------------------------------------------------

o================================================================================
|
|       p
|     p<        p          p_p         p
|   p<        p<_p       p<        p_p<      p p_p     p_p_p
| p<        p<         p<        p<        p<        p<
|
| 2^16      2^9        2^8       2^7       2^6       2^5
| 65536     512        256       128       64        32
|
o--------------------------------------------------------------------------------
|
|       p
|     p<        p          p_p         p
| p_p<      p_p<_p     p_p<      p_p_p<    p<        p_p_p_p
|                                            p p_p
|
| p_16      p_9        p_8       p_7       p_6       p_5
| 53        23         19        17        13        11
|
o--------------------------------------------------------------------------------
|
|   p^p       p_p        p         p
| p<        p<         p<        p<
|   p         p          p^p       p_p
|
| 3^4       3^3        7^2       5^2
| 81        27         49        25
|
o--------------------------------------------------------------------------------
|
|     p
| p p<      p p<       p^p p_p   p p_p_p
|     p         p^p
|
| 18        14         12        10
|
o================================================================================

Triangle in which k-th row lists natural number
values for the collection of riffs with k nodes.

k | natural numbers n such that |riff(n)| = k
--o------------------------------------------------
0 | 1;
1 | 2;
2 | 3, 4;
3 | 5, 6, 7, 8, 9, 16;
4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27,
  | 32, 49, 53, 64, 81, 128, 256, 512, 65536;

The natural number values for the riffs with
at most 3 pts are as follows (@'s are roots):

|                  o       o  o       o
|                  |       ^  |       ^
|                  v       |  v       |
|            o  o  o    o  o  o  o o  o
|            |  ^  |    |  |  ^  | ^  ^
|            v  |  v    v  v  |  v/   |
| Riff:   @; @, @; @, @ @, @, @, @,   @;
|
| Value:  2; 3, 4; 5,  6 , 7, 8, 9,  16;

---------------------------------------------------

1, 2, 3, 4, 5, 6, 7, 8, 9, 16,
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536,

---------------------------------------------------

1; 2; 3, 4; 5, 6, 7, 8, 9, 16;
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536;

---------------------------------------------------

Example of a Raw HTML Table

${\text{Table 10.}}~~{\text{Relation of Quantifiers to Higher Order Propositions}}$

${\text{Mnemonic}}\!$	${\text{Category}}\!$	${\text{Classical Form}}\!$	${\text{Alternate Form}}\!$	${\text{Symmetric Form}}\!$	${\text{Operator}}\!$
${\begin{matrix}\mathrm {E} \\\mathrm {Exclusive} \end{matrix}}$	${\begin{matrix}\mathrm {Universal} \\\mathrm {Negative} \end{matrix}}$	$\mathrm {All} ~u~\mathrm {is} ~{\texttt {(}}v{\texttt {)}}$		$\mathrm {No} ~u~\mathrm {is} ~v$	${\texttt {(}}\ell _{11}{\texttt {)}}$
${\begin{matrix}\mathrm {A} \\\mathrm {Absolute} \end{matrix}}$	${\begin{matrix}\mathrm {Universal} \\\mathrm {Affirmative} \end{matrix}}$	$\mathrm {All} ~u~\mathrm {is} ~v$		$\mathrm {No} ~u~\mathrm {is} ~{\texttt {(}}v{\texttt {)}}$	${\texttt {(}}\ell _{10}{\texttt {)}}$
		$\mathrm {All} ~v~\mathrm {is} ~u$	$\mathrm {No} ~v~\mathrm {is} ~{\texttt {(}}u{\texttt {)}}$	$\mathrm {No} ~{\texttt {(}}u{\texttt {)}}~\mathrm {is} ~v$	${\texttt {(}}\ell _{01}{\texttt {)}}$
		$\mathrm {All} ~{\texttt {(}}v{\texttt {)}}~\mathrm {is} ~u$	$\mathrm {No} ~{\texttt {(}}v{\texttt {)}}~\mathrm {is} ~{\texttt {(}}u{\texttt {)}}$	$\mathrm {No} ~{\texttt {(}}u{\texttt {)}}~\mathrm {is} ~{\texttt {(}}v{\texttt {)}}$	${\texttt {(}}\ell _{00}{\texttt {)}}$
		$\mathrm {Some} ~{\texttt {(}}u{\texttt {)}}~\mathrm {is} ~{\texttt {(}}v{\texttt {)}}$		$\mathrm {Some} ~{\texttt {(}}u{\texttt {)}}~\mathrm {is} ~{\texttt {(}}v{\texttt {)}}$	$\ell _{00}$
		$\mathrm {Some} ~{\texttt {(}}u{\texttt {)}}~\mathrm {is} ~v$		$\mathrm {Some} ~{\texttt {(}}u{\texttt {)}}~\mathrm {is} ~v$	$\ell _{01}$
${\begin{matrix}\mathrm {O} \\\mathrm {Obtrusive} \end{matrix}}$	${\begin{matrix}\mathrm {Particular} \\\mathrm {Negative} \end{matrix}}$	$\mathrm {Some} ~u~\mathrm {is} ~{\texttt {(}}v{\texttt {)}}$		$\mathrm {Some} ~u~\mathrm {is} ~{\texttt {(}}v{\texttt {)}}$	$\ell _{10}$
${\begin{matrix}\mathrm {I} \\\mathrm {Indefinite} \end{matrix}}$	${\begin{matrix}\mathrm {Particular} \\\mathrm {Affirmative} \end{matrix}}$	$\mathrm {Some} ~u~\mathrm {is} ~v$		$\mathrm {Some} ~u~\mathrm {is} ~v$	$\ell _{11}$

Figurate Numbers

Regular Polygonal Numbers

Data copied from Figurate numbers#Regular polygonal numbers

Wiki Table

–(A067998)	0-gonal numbers	$-n^{2}+2n\!$	1	0	–3	–8	–15	–24	–35	–48	–63	–80	–99	–120	–143	–168	–195	–224	–255	–288	–323	–360
A080956	1-gonal numbers	${\frac {-n^{2}+3n}{2}}\!$	1	1	0	–2	–5	–9	–14	–20	–27	–35	–44	–54	–65	–77	–90	–104	–119	–135	–152	–170
A000027	2-gonal numbers	$n\!$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
A000217	Triangular numbers	${\frac {n^{2}+n}{2}}\!$	1	3	6	10	15	21	28	36	45	55	66	78	91	105	120	136	153	171	190	210
A000290	Square numbers	$n^{2}\!$	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225	256	289	324	361	400
A000326	Pentagonal numbers	${\frac {3n^{2}-n}{2}}\!$	1	5	12	22	35	51	70	92	117	145	176	210	247	287	330	376	425	477	532	590
A000384	Hexagonal numbers	$2n^{2}-n\!$	1	6	15	28	45	66	91	120	153	190	231	276	325	378	435	496	561	630	703	780
A000566	Heptagonal numbers	${\frac {5n^{2}-3n}{2}}\!$	1	7	18	34	55	81	112	148	189	235	286	342	403	469	540	616	697	783	874	970
A000567	Octagonal numbers	$3n^{2}-2n\!$	1	8	21	40	65	96	133	176	225	280	341	408	481	560	645	736	833	936	1045	1160
A001106	Nonagonal numbers	${\frac {7n^{2}-5n}{2}}\!$	1	9	24	46	75	111	154	204	261	325	396	474	559	651	750	856	969	1089	1216	1350
A001107	Decagonal numbers	$4n^{2}-3n\!$	1	10	27	52	85	126	175	232	297	370	451	540	637	742	855	976	1105	1242	1387	1540
A051682	11-gonal numbers	${\frac {9n^{2}-7n}{2}}\!$	1	11	30	58	95	141	196	260	333	415	506	606	715	833	960	1096	1241	1395	1558	1730
A051624	12-gonal numbers	$5n^{2}-4n\!$	1	12	33	64	105	156	217	288	369	460	561	672	793	924	1065	1216	1377	1548	1729	1920
A051865	13-gonal numbers	${\frac {11n^{2}-9n}{2}}\!$	1	13	36	70	115	171	238	316	405	505	616	738	871	1015	1170	1336	1513	1701	1900	2110
A051866	14-gonal numbers	$6n^{2}-5n\!$	1	14	39	76	125	186	259	344	441	550	671	804	949	1106	1275	1456	1649	1854	2071	2300
A051867	15-gonal numbers	${\frac {13n^{2}-11n}{2}}\!$	1	15	42	82	135	201	280	372	477	595	726	870	1027	1197	1380	1576	1785	2007	2242	2490
A051868	16-gonal numbers	$7n^{2}-6n\!$	1	16	45	88	145	216	301	400	513	640	781	936	1105	1288	1485	1696	1921	2160	2413	2680
A051869	17-gonal numbers	${\frac {15n^{2}-13n}{2}}\!$	1	17	48	94	155	231	322	428	549	685	836	1002	1183	1379	1590	1816	2057	2313	2584	2870
A051870	18-gonal numbers	$8n^{2}-7n\!$	1	18	51	100	165	246	343	456	585	730	891	1068	1261	1470	1695	1936	2193	2466	2755	3060
A051871	19-gonal numbers	${\frac {17n^{2}-15n}{2}}\!$	1	19	54	106	175	261	364	484	621	775	946	1134	1339	1561	1800	2056	2329	2619	2926	3250
A051872	20-gonal numbers	$9n^{2}-8n\!$	1	20	57	112	185	276	385	512	657	820	1001	1200	1417	1652	1905	2176	2465	2772	3097	3440

Centered Polygonal Numbers

Data copied from Figurate numbers#Centered polygonal numbers

Wiki Table

A000012	Centered 0-gonal numbers	$1\!$	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
A000124	Centered 1-gonal numbers	$T_{n-1}+1\!$	1	2	4	7	11	16	22	29	37	46	56	67	79	92	106	121	137	154	172	191
A002061	Centered 2-gonal numbers	$2T_{n-1}+1\!$	1	3	7	13	21	31	43	57	73	91	111	133	157	183	211	241	273	307	343	381
A005448	Centered triangular numbers	$3T_{n-1}+1\!$	1	4	10	19	31	46	64	85	109	136	166	199	235	274	316	361	409	460	514	571
A001844	Centered square numbers	$4T_{n-1}+1\!$	1	5	13	25	41	61	85	113	145	181	221	265	313	365	421	481	545	613	685	761
A005891	Centered pentagonal numbers	$5T_{n-1}+1\!$	1	6	16	31	51	76	106	141	181	226	276	331	391	456	526	601	681	766	856	951
A003215	Centered hexagonal numbers	$6T_{n-1}+1\!$	1	7	19	37	61	91	127	169	217	271	331	397	469	547	631	721	817	919	1027	1141
A069099	Centered heptagonal numbers	$7T_{n-1}+1\!$	1	8	22	43	71	106	148	197	253	316	386	463	547	638	736	841	953	1072	1198	1331
A016754	Centered octagonal numbers	$(2n+1)^{2}\!$	1	9	25	49	81	121	169	225	289	361	441	529	625	729	841	961	1089	1225	1369	1521
A060544	Centered nonagonal numbers	$T_{3(n-1)+1}\!$	1	10	28	55	91	136	190	253	325	406	496	595	703	820	946	1081	1225	1378	1540	1711
A062786	Centered decagonal numbers	$10T_{n-1}+1\!$	1	11	31	61	101	151	211	281	361	451	551	661	781	911	1051	1201	1361	1531	1711	1901
A069125	Centered 11-gonal numbers	$11T_{n-1}+1\!$	1	12	34	67	111	166	232	309	397	496	606	727	859	1002	1156	1321	1497	1684	1882	2091
A003154	Centered 12-gonal numbers	$12T_{n-1}+1\!$	1	13	37	73	121	181	253	337	433	541	661	793	937	1093	1261	1441	1633	1837	2053	2281
A069126	Centered 13-gonal numbers	$13T_{n-1}+1\!$	1	14	40	79	131	196	274	365	469	586	716	859	1015	1184	1366	1561	1769	1990	2224	2471
A069127	Centered 14-gonal numbers	$14T_{n-1}+1\!$	1	15	43	85	141	211	295	393	505	631	771	925	1093	1275	1471	1681	1905	2143	2395	2661
A069128	Centered 15-gonal numbers	$15T_{n-1}+1\!$	1	16	46	91	151	226	316	421	541	676	826	991	1171	1366	1576	1801	2041	2296	2566	2851
A069129	Centered 16-gonal numbers	$16T_{n-1}+1\!$	1	17	49	97	161	241	337	449	577	721	881	1057	1249	1457	1681	1921	2177	2449	2737	3041
A069130	Centered 17-gonal numbers	$17T_{n-1}+1\!$	1	18	52	103	171	256	358	477	613	766	936	1123	1327	1548	1786	2041	2313	2602	2908	3231
A069131	Centered 18-gonal numbers	$18T_{n-1}+1\!$	1	19	55	109	181	271	379	505	649	811	991	1189	1405	1639	1891	2161	2449	2755	3079	3421
A069132	Centered 19-gonal numbers	$19T_{n-1}+1\!$	1	20	58	115	191	286	400	533	685	856	1046	1255	1483	1730	1996	2281	2585	2908	3250	3611
A069133	Centered 20-gonal numbers	$20T_{n-1}+1\!$	1	21	61	121	201	301	421	561	721	901	1101	1321	1561	1821	2101	2401	2721	3061	3421	3801