View source for Sequence Sage

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{{DISPLAYTITLE:Sequence ''Sage''}}
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<math>\text{Sage}(n) = \frac{  2^{2^n}  }{  2^{n{\color{Red}+1}}  } ~=~ 2^{(2^n - n {\color{Red}- 1})}</math> {{spaces|20}}
<math>\text{DubSage}(n) = \frac{2^{2^n}}{2^n} ~=~ 2^{(2^n - n)}</math>

==Sequences ''Sage<sub>(Drop)</sub>''==

{| class="wikitable" style="text-align: center;"
!
! 0 !!  1  !!  2  !!  3  !!  4  !!  5  !!  6 !!  7
|- style="font-size: 70%;"
! Eulerian numbers <small>({{oeis|A000295}})</small>
| 0 || 0 || 1 || 4 || 11 || 26 || 57 || 120
|-
! 🌊 ''Sage'' <small>(~{{oeis|A016031}})</small>
| 1 || 1 || 2 || 16 || 2048 || 67108864 || 144115188075855872 || 1329227995784915872903807060280344576
|- 
! 💧 ''SageDrop''
| 1 || 0 || 1 || 14 || 2032 || 67106816 || 144115188008747008 || 1329227995784915872759691872204488704
|}

{| class="boolf-seq-legend"
|class="nutshell"| ''a'' ↦ sharp families
|class="left"| Sage(''a'')<br><span class="drop">SageDrop(''a'')</span>
| is the number of {{boolf-prop|sharp}} {{boolf-prop|family|families}} with
| {{boolf-prop|arity}}<br><span class="drop">{{boolf-prop|adicity}}</span>
|class="right"| ''a''.
|}

=={{anchor|dub}}Sequences ''DubSage<sub>(Drop)</sub>''==

{| class="wikitable" style="text-align: center;"
!
! 0 !!  1  !!  2  !!  3  !!  4  !!  5  !!  6 !!  7
|-
! 🌊 ''DubSage''
| 2 || 2 || 4 || 32 || 4096 || 134217728 || 288230376151711744 || 2658455991569831745807614120560689152
|- 
! 💧 ''DubSageDrop''
| 2 || 0 || 2 || 28 || 4064 || 134213632 || 288230376017494016 || 2658455991569831745519383744408977408
|}

{| class="boolf-seq-legend"
|class="nutshell"| ''a'' ↦ extroverted families
|class="left"| DubSage(''a'')<br><span class="drop">DubSageDrop(''a'')</span>
| is the number of {{boolf-prop|extroverted}} {{boolf-prop|family|families}} with
| {{boolf-prop|arity}}<br><span class="drop">{{boolf-prop|adicity}}</span>
|class="right"| ''a''.
|}

=={{anchor|torn}}Sequences ''TornSage<sub>(Drop)</sub>''==

<math>\text{for} ~ n \ge 2: ~ \text{TornSage}(n) = \frac{\text{Sage}(n)}{2}</math>

{| class="wikitable" style="text-align: center;"
!
!  1  !!  2  !!  3  !!  4  !!  5  !!  6 !!  7
|-
! 🌊 ''TornSage'' <small>(~{{oeis|A076688}})</small>
| 1 || 1 || 8 || 1024 || 33554432 || 72057594037927936 || 664613997892457936451903530140172288
|- 
! 💧 ''TornSageDrop''
| 1 || 0 || 7 || 1016 || 33553408 || 72057594004373504 || 664613997892457936379845936102244352
|}

{| class="boolf-seq-legend"
|class="nutshell"| ''a'' ↦ sharp super-families
|class="left"| TornSage(''a'')<br><span class="drop">TornSageDrop(''a'')</span>
| is the number of {{boolf-prop|sharp}} {{boolf-prop|super-family|super-families}} with
| {{boolf-prop|arity}}<br><span class="drop">{{boolf-prop|adicity}}</span>
|class="right"| ''a''.
|}

Compare {{boolf-seq|Clove}}.


[[Category:Boolf sequences/pages]]