User:Jon Awbrey/SEQUENCES
A061396
Plain Wiki Table
| \(\text{Integer}\!\) | \(\text{Factorization}\!\) | \(\text{Notation}\!\) | \(\text{Riff Digraph}\!\) | \(\text{Rote Graph}\!\) | \(\text{Traversal}\!\) |
| \(1\!\) | \(1\!\) | 
|
|||
| \(2\!\) | \(\text{p}_1^1\!\) | \(\text{p}\!\) |
|
|
\(((~))\) |
| \(3\!\) |
\(\begin{array}{lll} \text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1 \end{array}\) |
\(\text{p}_\text{p}\!\) | \(\cdots\) | 
|
\((((~))(~))\) |
| \(4\!\) |
\(\begin{array}{lll} \text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1} \end{array}\) |
\(\text{p}^\text{p}\!\) | \(\cdots\) | 
|
\(((((~))))\) |
| \(5\!\) |
\(\begin{array}{lll} \text{p}_3^1 & = & \text{p}_{\text{p}_2^1}^1 \\[6pt] & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 \end{array}\) |
\(\text{p}_{\text{p}_{\text{p}}}\!\) | \(\cdots\) | 
|
\(((((~))(~))(~))\) |
| \(6\!\) |
\(\begin{array}{lll} \text{p}_1^1 \text{p}_2^1 & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1 \end{array}\) |
\(\text{p} \text{p}_{\text{p}}\!\) | \(\cdots\) | 
|
\(((~))(((~))(~))\) |
| \(7\!\) |
\(\begin{array}{lll} \text{p}_4^1 & = & \text{p}_{\text{p}_1^2}^1 \\[6pt] & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 \end{array}\) |
\(\text{p}_{\text{p}^{\text{p}}}\!\) | \(\cdots\) | 
|
\((((((~))))(~))\) |
| \(8\!\) |
\(\begin{array}{lll} \text{p}_1^3 & = & \text{p}_1^{\text{p}_2^1} \\[6pt] & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} \end{array}\) |
\(\text{p}^{\text{p}_{\text{p}}}\!\) | \(\cdots\) | 
|
\((((((~))(~))))\) |
| \(9\!\) |
\(\begin{array}{lll} \text{p}_2^2 & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1} \end{array}\) |
\(\text{p}_\text{p}^\text{p}\!\) | \(\cdots\) | 
|
\((((~))(((~))))\) |
| \(16\!\) |
\(\begin{array}{lll} \text{p}_1^4 & = & \text{p}_1^{\text{p}_1^2} \\[6pt] & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} \end{array}\) |
\(\text{p}^{\text{p}^{\text{p}}}\!\) | \(\cdots\) | 
|
\(((((((~))))))\) |
Nested Wiki Table
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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
| \(\text{Mnemonic}\!\) | \(\text{Category}\!\) | \(\text{Classical Form}\!\) | \(\text{Alternate Form}\!\) | \(\text{Symmetric Form}\!\) | \(\text{Operator}\!\) |
| \(\begin{matrix} \mathrm{E} \\ \mathrm{Exclusive} \end{matrix}\) | \(\begin{matrix} \mathrm{Universal} \\ \mathrm{Negative} \end{matrix}\) | \(\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\mathrm{No} ~ u ~ \mathrm{is} ~ v\) | \(\texttt{(} \ell_{11} \texttt{)}\) | |
| \(\begin{matrix} \mathrm{A} \\ \mathrm{Absolute} \end{matrix}\) | \(\begin{matrix} \mathrm{Universal} \\ \mathrm{Affirmative} \end{matrix}\) | \(\mathrm{All} ~ u ~ \mathrm{is} ~ v\) | \(\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\texttt{(} \ell_{10} \texttt{)}\) | |
| \(\mathrm{All} ~ v ~ \mathrm{is} ~ u\) | \(\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}\) | \(\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\) | \(\texttt{(} \ell_{01} \texttt{)}\) | ||
| \(\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u\) | \(\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}\) | \(\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\texttt{(} \ell_{00} \texttt{)}\) | ||
| \(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\ell_{00}\) | |||
| \(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\) | \(\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v\) | \(\ell_{01}\) | |||
| \(\begin{matrix} \mathrm{O} \\ \mathrm{Obtrusive} \end{matrix}\) | \(\begin{matrix} \mathrm{Particular} \\ \mathrm{Negative} \end{matrix}\) | \(\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}\) | \(\ell_{10}\) | |
| \(\begin{matrix} \mathrm{I} \\ \mathrm{Indefinite} \end{matrix}\) | \(\begin{matrix} \mathrm{Particular} \\ \mathrm{Affirmative} \end{matrix}\) | \(\mathrm{Some} ~ u ~ \mathrm{is} ~ v\) | \(\mathrm{Some} ~ u ~ \mathrm{is} ~ v\) | \(\ell_{11}\) |