Difference between revisions of "Directory:Jon Awbrey/Papers/Riffs and Rotes"
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==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> | ||
| + | |} | ||
Revision as of 14:28, 19 January 2010
Riffs in Numerical Order
|
\(\begin{array}{l} \varnothing \\ 1 \end{array}\) |
\(\text{p}\!\) \(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\) |
\(\text{p}_\text{p}\!\) \(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\) |
\(\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\) |
\(\text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\) |
|
\(\text{p} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\) |
\(\text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\) |
\(\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\) |
\(\text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\) |
|
\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\) |
\(\text{p}^\text{p} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\) |
\(\text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\) |
\(\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\) |
|
\(\text{p}^{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\) |
\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\) |
\(\text{p} \text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\) |
\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\) |
|
\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\) |
\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\) |
\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\) |
|
\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\) |
\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\) |
\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\) |
|
\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\) \(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\) |
\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\) |
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\) |
\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\) |
|
\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\) |
\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\) |
\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\) |
\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\) |
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\) |
|
\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\) \(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\) |
\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\) |
\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\) |
\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\) |
\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\) |
|
\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\) |
\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\) |
\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\) \(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\) |
\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\) \(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\) |
\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\) |
|
\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\) |
\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\) |
\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\) \(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\) |
\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\) |
\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\) |
|
\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\) \(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\) |
\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\) |
\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\) \(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\) |
\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\) \(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\) |
\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\) \(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\) |
Rotes in Numerical Order
|
\(\begin{array}{l} \varnothing \\ 1 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\) |
\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\) |
\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\) |
\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\) |
\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\) |
|
\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\) |
\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\) |
\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\) |
\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\) |
|
\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\) |
\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\) |
\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\) |
\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\) |
\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\) |
\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\) |
|
\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\) |
\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\) |
\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\) |
\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\) |
\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\) |
|
\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\) |
\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\) |
\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\) |
\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\) |
\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\) |
\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\) |
|
\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\) |
\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\) |
\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\) |
|
\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\) |
\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\) |
\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\) |
\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\) |
\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\) |
Selected Sequences
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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.
A062504
- Triangle in which k-th row lists natural number values for the collection of riffs with k nodes.
|
\(\begin{array}{l|l|r} k & P_k = \{ n : \operatorname{riff}(n) ~\text{has}~ k ~\text{nodes} \} = \{ n : \operatorname{rote}(n) ~\text{has}~ 2k + 1 ~\text{nodes} \} & |P_k| \\[10pt] 0 & \{ 1 \} & 1 \\ 1 & \{ 2 \} & 1 \\ 2 & \{ 3, 4 \} & 2 \\ 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 \\ 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 \end{array}\) |
