Triangle ExOlive

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

Triangle Oak with multiplied columns

Triangles ExOlive_(Drop)

ExOlive = Olive ∘ powers of two

Oak with columns multiplied by ExAloe

🌊 triangle ExOlive row sums Grass
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	1 · 2 2	1 · 2 2					4
2	1 · 2 2	3 · 2 6	1 · 8 8				16
3	1 · 2 2	7 · 2 14	7 · 8 56	1 · 184 184			256
4	1 · 2 2	15 · 2 30	35 · 8 280	15 · 184 2760	1 · 62464 62464		65536
5	1 · 2 2	31 · 2 62	155 · 8 1240	155 · 184 28520	31 · 62464 1936384	1 · 4293001088 4293001088	4294967296

OakDrop with columns multiplied by ExAloe

💧 triangle ExOliveDrop row sums GrassDrop
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	0	1 · 2 2					2
2	0	2 · 2 4	1 · 8 8				12
3	0	4 · 2 8	6 · 8 48	1 · 184 184			240
4	0	8 · 2 16	28 · 8 224	14 · 184 2576	1 · 62464 62464		65280
5	0	16 · 2 32	120 · 8 960	140 · 184 25760	30 · 62464 1873920	1 · 4293001088 4293001088	4294901760

2-padded Forsythia with rows multiplied by powers of two.

💧 triangle ExOliveDrop row sums GrassDrop
k n	0	1	2	3	4	5	sums
0	2 · 1 2						2
1	0	1 · 2 2					2
2	0	1 · 4 4	2 · 4 8				12
3	0	1 · 8 8	6 · 8 48	23 · 8 184			240
4	0	1 · 16 16	14 · 16 224	161 · 16 2576	3904 · 16 62464		65280
5	0	1 · 32 32	30 · 32 960	805 · 32 25760	58560 · 32 1873920	134156284 · 32 4293001088	4294901760

(a, s) ↦ BF

ExOlive(a, s)
ExOliveDrop(a, s)

is the number of BF with

a and strength s.

Column 0 is ExAloe_(Drop). The row sums without it are NonExAloe_(Drop).

🌊 triangle TurnedExOlive row sums Grass
k n	0	1	2	3	4	5	sums
0	2						2
1	2	2					4
2	8	6	2				16
3	184	56	14	2			256
4	62464	2760	280	30	2		65536
5	4293001088	1936384	28520	1240	62	2	4294967296

💧 triangle TurnedExOliveDrop row sums GrassDrop
k n	0	1	2	3	4	5	sums
0	2						2
1	0	2					2
2	6	4	2				12
3	176	50	12	2			240
4	62280	2704	266	28	2		65280
5	4292938624	1933624	28240	1210	60	2	4294901760

(a, d) ↦ BF

TurnedExOlive(a, d)
TurnedExOliveDrop(a, d)

is the number of BF with

🌊 triangle SmallTurnedExOlive row sums SmallGrass
k n	0	1	2	3	4	5	sums
0	1						1
1	2	1					3
2	8	6	1				15
3	184	56	14	1			255
4	62464	2760	280	30	1		65535
5	4293001088	1936384	28520	1240	62	1	4294967295

💧 triangle SmallTurnedExOliveDrop row sums SmallGrassDrop
k n	0	1	2	3	4	5	sums
0	1						1
1	1	1					2
2	6	5	1				12
3	176	50	13	1			240
4	62280	2704	266	29	1		65280
5	4292938624	1933624	28240	1210	61	1	4294901760

(a, d) ↦ BF

SmallTurnedExOlive(a, d)
SmallTurnedExOliveDrop(a, d)

is the number of BF with

a and depth d. (The contradiction with infinite depth is not counted.).