12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Thursday November 28, 2024
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Riffs and Rotes (view source)
Revision as of 14:28, 19 January 2010
, 14:28, 19 January 2010→Selected Sequences
==Selected Sequences==
==Selected Sequences==
{| align="center" border="1" width="90%"
|+ style="height:25px" | <math>\text{Prime Factorizations, Riffs, and Rotes}\!</math>
|- style="height:50px; background:#f0f0ff"
|
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
| width="10%" | <math>\text{Integer}\!</math>
| width="25%" | <math>\text{Factorization}\!</math>
| width="15%" | <math>\text{Notation}\!</math>
| width="25%" | <math>\text{Riff Digraph}\!</math>
| width="25%" | <math>\text{Rote Graph}\!</math>
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>1\!</math>
| width="25%" | <math>1\!</math>
| width="15%" |
| width="25%" |
| width="25%" | [[Image:Rote 1 Big.jpg|20px]]
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>2\!</math>
| width="25%" | <math>\text{p}_1^1\!</math>
| width="15%" | <math>\text{p}\!</math>
| width="25%" | [[Image:Riff 2 Big.jpg|20px]]
| width="25%" | [[Image:Rote 2 Big.jpg|40px]]
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>3\!</math>
| width="25%" |
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| width="15%" | <math>\text{p}_\text{p}\!</math>
| width="25%" | [[Image:Riff 3 Big.jpg|40px]]
| width="25%" | [[Image:Rote 3 Big.jpg|40px]]
|-
| <math>4\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| [[Image:Riff 4 Big.jpg|40px]]
| [[Image:Rote 4 Big.jpg|65px]]
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>5\!</math>
| width="25%" |
<math>\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}</math>
| width="15%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="25%" | [[Image:Riff 5 Big.jpg|65px]]
| width="25%" | [[Image:Rote 5 Big.jpg|40px]]
|-
| <math>6\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| [[Image:Riff 6 Big.jpg|65px]]
| [[Image:Rote 6 Big.jpg|80px]]
|-
| <math>7\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 7 Big.jpg|65px]]
| [[Image:Rote 7 Big.jpg|65px]]
|-
| <math>8\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| [[Image:Riff 8 Big.jpg|65px]]
| [[Image:Rote 8 Big.jpg|65px]]
|-
| <math>9\!</math>
|
<math>\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| [[Image:Riff 9 Big.jpg|40px]]
| [[Image:Rote 9 Big.jpg|80px]]
|-
| <math>16\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| [[Image:Riff 16 Big.jpg|65px]]
| [[Image:Rote 16 Big.jpg|90px]]
|}
|}
===A061396===
* '''Number of "rooted index-functional forests" (Riffs) on n nodes.'''
* '''Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.'''
* [http://oeis.org/wiki/A061396 OEIS Wiki Entry for A061396].
===A062504===
* '''Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.'''
* [http://oeis.org/wiki/A062504 OEIS Wiki Entry for A062504].
{| align="center"
|
<math>\begin{array}{l|l|r}
k
& P_k
= \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \}
= \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \}
& |P_k|
\\[10pt]
0 & \{ 1 \} & 1
\\
1 & \{ 2 \} & 1
\\
2 & \{ 3, 4 \} & 2
\\
3 & \{ 5, 6, 7, 8, 9, 16 \} & 6
\\
4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20
\end{array}</math>
|}
