Triangle StackedHeather

Studies of Boolean functions
sequences related to seals

By adding a dimension for depth, these triangles can be refined to pyramids StackedLime etc.

Triangles StackedHeather and DiagonalHeather

StackedHeather(a, v) = Heather(v), iff v ≤ a.
DiagonalHeather(a, v) = Heather(v), iff v = a.

🌊    triangle StackedHeather    row sums Clover
v
a
 0 1 2 3 4 5 6 7 sums
0 1 1
1 1 1 2
2 1 1 2 4
3 1 1 2 4 8
4 1 1 2 4 8 16
5 1 1 2 4 8 16 32
6 1 1 2 4 8 16 36 68
7 1 1 2 4 8 16 36 80 148
💧    triangle DiagonalHeather    row sums Heather
v
a
 0 1 2 3 4 5 6 7 sums
0 1 1
1 0 1 1
2 0 0 2 2
3 0 0 0 4 4
4 0 0 0 0 8 8
5 0 0 0 0 0 16 16
6 0 0 0 0 0 0 36 36
7 0 0 0 0 0 0 0 80 80
(a, v) ↦ houses StackedHeather(a, v)
DiagonalHeather(a, v) 		is the number of houses of the seals with arity
adicity 		a and valency v.

Compare MaimedStackedHeather.

Triangles SlopedHeather and SlopedLavender

SlopedHeather(a, g) = Heather(a−g), iff g ≤ a.
SlopedLavender(a, g) = Lavender(a−g), iff g ≤ a.

🌊    triangle SlopedHeather    row sums Clover
g
a
 0 1 2 3 4 5 6 7 sums
0 1 1
1 1 1 2
2 2 1 1 4
3 4 2 1 1 8
4 8 4 2 1 1 16
5 16 8 4 2 1 1 32
6 36 16 8 4 2 1 1 68
7 80 36 16 8 4 2 1 1 148
💧    triangle SlopedLavender    row sums Heather
g
a
 0 1 2 3 4 5 6 7 sums
0 1 1
1 0 1 1
2 1 0 1 2
3 2 1 0 1 4
4 4 2 1 0 1 8
5 8 4 2 1 0 1 16
6 20 8 4 2 1 0 1 36
7 44 20 8 4 2 1 0 1 80
(a, g) ↦ houses SlopedHeather(a, g)
SlopedLavender(a, g) 		is the number of houses of the seals with arity
adicity 		a and gravity g.
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