Sequence Sage

Studies of Boolean functions
sequences related to Boolean functions

families
closed: NonDubSage	blunt: NonSage
unclosed: DubSage	sharp: Sage
total: Thyme
super-families
closed: NonDubSage	blunt: Clove
unclosed: Sage	sharp: TornSage
total: NonSage

${\text{Sage}}(n)={\frac {2^{2^{n}}}{2^{n{\color {Red}+1}}}}~=~2^{(2^{n}-n{\color {Red}-1})}$ ${\text{DubSage}}(n)={\frac {2^{2^{n}}}{2^{n}}}~=~2^{(2^{n}-n)}$

Sequences DubSage_(Drop)

	0	1	2	3	4	5	6	7
2ⁿ − n (A000325)	1	1	2	5	12	27	58	121
🌊 DubSage (A300361)	2	2	4	32	4096	134217728	288230376151711744	2658455991569831745807614120560689152
💧 DubSageDrop	2	0	2	28	4064	134213632	288230376017494016	2658455991569831745519383744408977408

a ↦ unclosed families

DubSage(a)
DubSageDrop(a)

is the number of unclosed families with

arity
adicity

a.

Sequences Sage_(Drop)

	0	1	2	3	4	5	6	7
2ⁿ − n − 1 (A000295)	0	0	1	4	11	26	57	120
🌊 Sage (~A016031)	1	1	2	16	2048	67108864	144115188075855872	1329227995784915872903807060280344576
💧 SageDrop	1	0	1	14	2032	67106816	144115188008747008	1329227995784915872759691872204488704

a ↦ sharp families

Sage(a)
SageDrop(a)

is the number of sharp families with

arity
adicity

a.

Sequences TornSage_(Drop)

${\text{for}}~n\geq 2:~{\text{TornSage}}(n)={\frac {{\text{Sage}}(n)}{2}}$

	0	1	2	3	4	5	6	7
🌊 TornSage (~A076688)	0	1	1	8	1024	33554432	72057594037927936	664613997892457936451903530140172288
💧 TornSageDrop	0	1	0	7	1016	33553408	72057594004373504	664613997892457936379845936102244352

a ↦ sharp super-families

TornSage(a)
TornSageDrop(a)

is the number of sharp super-families with

arity
adicity

a.

Sequences DubSage(Drop)

Sequences Sage(Drop)

Sequences TornSage(Drop)

Sequences DubSage_(Drop)

Sequences Sage_(Drop)

Sequences TornSage_(Drop)