Changes

MyWikiBiz, Author Your Legacy — Sunday April 06, 2025
Jump to navigationJump to search

User:Jon Awbrey/SEQUENCES (view source)

Revision as of 06:28, 31 December 2009

4,503 bytes added ,  06:28, 31 December 2009
no edit summary
Line 1,013: Line 1,013:     
---------------------------------------------------
 
---------------------------------------------------
 +
</pre>
 +
 +
==A062504==
 +
 +
* [http://oeis.org/wiki/A062504 A062504]
 +
 +
===TeX Array===
 +
 +
{| 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>
 +
|}
 +
 +
===JPEG===
 +
 +
{| 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%" | &nbsp;
 +
| width="25%" | &nbsp;
 +
| 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]]
 +
|}
 +
|}
 +
 +
===ASCII===
 +
 +
<pre>
 +
Example
 +
 +
    * k | natural numbers n such that |riff(n)| = k
 +
    * 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 (x = root):
 +
    * .................o.......o..o.......o
 +
    * .................|.......^..|.......^
 +
    * .................v.......|..v.......|
 +
    * ...........o..o..o....o..o..o..o.o..o
 +
    * ...........|..^..|....|..|..^..|.^..^
 +
    * ...........v..|..v....v..v..|..v/...|
 +
    * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
 +
    * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;
 
</pre>
 
</pre>
  
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/103204"

Navigation menu