Triangle ExOlive

Studies of Boolean functions
sequences related to Boolean functions
triangles by arity and strength
base
triangle 		diagonal families BF
triangle total triangle total
all Oak (Ex)Aloe Olive Thyme ExOlive Grass
balanced (Ex)Agave Laurel Saffron ExLaurel Sesame
dense Birch (Ex)Aloe Lemon Lily ExLemon Lotus
b & d (Ex)Agave Orange Chamomile ExOrange Calendula
valency
gravity 		strength
depth
triangle Willow ExOlive
diagonal Lotus ExAloe
difference NonLotus NonExAloe
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 arity
adicity 		a and strength s.

Triangles TurnedExOlive(Drop)

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 arity
adicity 		a and soft depth d.

Triangles SmallTurnedExOlive(Drop)

Column 0 is SmallExAloe(Drop).   The row sums without it are SmallNonExAloe(Drop).

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