Sequence ExAloe

Studies of Boolean functions
sequences related to Boolean functions

triangles by arity and strength
	base triangle	diagonal	families		BF
	base triangle	diagonal	triangle	total	triangle	total
all	Oak	(Ex)Aloe	Olive	Thyme	ExOlive	Grass
balanced	Oak	(Ex)Agave	Laurel	Saffron	ExLaurel	Sesame
dense	Birch	(Ex)Aloe	Lemon	Lily	ExLemon	Lotus
b & d	Birch	(Ex)Agave	Orange	Chamomile	ExOrange	Calendula

Sequences ExAloe_(Drop)


	0	1	2	3	4	5	6	7
🌊 ExAloe	2	2	8	184	62464	4293001088	18446743803209556992	340282366920938461120638132973980614656
💧 ExAloeDrop	2	0	6	176	62280	4292938624	18446743798916555904	340282366920938461102191389170771057664

a ↦ strong

ExAloe(a) is the number of strong BF with arity a.

a ↦ depth 0

ExAloe(a)
ExAloeDrop(a)

is the number of BF with

ExAloe is the product of Aloe and the powers of two.

${\text{ExAloe}}(n)={\color {Green}{\text{Aloe}}(n)}\cdot {\color {Red}2^{n}}$

row sums of SignedBramble

The absolute values are Oak with diagonals multiplied by Chrysanthemum and columns multiplied by Grass.

triangle SignedBramble row sums ExAloe
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 · 2 2								2
1	1 · 1 · 2 −2	1 · 1 · 4 4							2
2	1 · 2 · 2 4	3 · 1 · 4 −12	1 · 1 · 16 16						8
3	1 · 8 · 2 −16	7 · 2 · 4 56	7 · 1 · 16 −112	1 · 1 · 256 256					184
4	1 · 64 · 2 128	15 · 8 · 4 −480	35 · 2 · 16 1120	15 · 1 · 256 −3840	1 · 1 · 65536 65536				62464
5	1 · 1024 · 2 −2048	31 · 64 · 4 7936	155 · 8 · 16 −19840	155 · 2 · 256 79360	31 · 1 · 65536 −2031616	1 · 1 · 4294967296 4294967296			4293001088
6	1 · 32768 · 2 65536	63 · 1024 · 4 −258048	651 · 64 · 16 666624	1395 · 8 · 256 −2856960	651 · 2 · 65536 85327872	63 · 1 · 4294967296 −270582939648	1 · 1 · 18446744073709551616 18446744073709551616		18446743803209556992
7	1 · 2097152 · 2 −4194304	127 · 32768 · 4 16646144	2667 · 1024 · 16 −43696128	11811 · 64 · 256 193511424	11811 · 8 · 65536 −6192365568	2667 · 2 · 4294967296 22909355556864	127 · 1 · 18446744073709551616 −2342736497361113055232	1 · 1 · 340282366920938463463374607431768211456 340282366920938463463374607431768211456	340282366920938461120638132973980614656

SmallExAloe differs from ExAloe only in the first entry.
SmallExAloe_(Drop) are the left columns of SmallTurnedExOlive_(Drop).


	0	1	2	3	4
🌊 SmallExAloe (A305737)	1	2	8	184	62464
💧 SmallExAloeDrop	1	1	6	176	62280

a ↦ depth 0

SmallExAloe(a)
SmallExAloeDrop(a)

is the number of BF with

a and depth 0.

row sums of SignedSmallBramble

The absolute values are Oak with diagonals multiplied by Chrysanthemum and columns multiplied by SmallGrass.

SignedSmallBramble row sums SmallExAloe (A305737)
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 · 1 1								1
1	1 · 1 · 1 −1	1 · 1 · 3 3							2
2	1 · 2 · 1 2	3 · 1 · 3 −9	1 · 1 · 15 15						8
3	1 · 8 · 1 −8	7 · 2 · 3 42	7 · 1 · 15 −105	1 · 1 · 255 255					184
4	1 · 64 · 1 64	15 · 8 · 3 −360	35 · 2 · 15 1050	15 · 1 · 255 −3825	1 · 1 · 65535 65535				62464
5	1 · 1024 · 1 −1024	31 · 64 · 3 5952	155 · 8 · 15 −18600	155 · 2 · 255 79050	31 · 1 · 65535 −2031585	1 · 1 · 4294967295 4294967295			4293001088
6	1 · 32768 · 1 32768	63 · 1024 · 3 −193536	651 · 64 · 15 624960	1395 · 8 · 255 −2845800	651 · 2 · 65535 85326570	63 · 1 · 4294967295 −270582939585	1 · 1 · 18446744073709551615 18446744073709551615		18446743803209556992
7	1 · 2097152 · 1 −2097152	127 · 32768 · 3 12484608	2667 · 1024 · 15 −40965120	11811 · 64 · 255 192755520	11811 · 8 · 65535 −6192271080	2667 · 2 · 4294967295 22909355551530	127 · 1 · 18446744073709551615 −2342736497361113055105	1 · 1 · 340282366920938463463374607431768211455 340282366920938463463374607431768211455	340282366920938461120638132973980614656