Triangle Cedar

Studies of Boolean functions
sequences related to Boolean functions

gender
	diagonal	triangle	total
♀	Violet	Robinia	Primula
♂	HalfGrass	HalfCedar	HalfCrocus
all	Lotus	Willow	Grass
♂ − ♀	NonLotus	Hickory	Hyacinth

Pascal's triangle with multiplied columns

Triangles Cedar_(Drop)

The diagonal of both triangles is Grass.

Pascal with columns multiplied by Grass

🌊 triangle Cedar row sums Crocus
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	1 · 2 2	1 · 4 4					6
2	1 · 2 2	2 · 4 8	1 · 16 16				26
3	1 · 2 2	3 · 4 12	3 · 16 48	1 · 256 256			318
4	1 · 2 2	4 · 4 16	6 · 16 96	4 · 256 1024	1 · 65536 65536		66674
5	1 · 2 2	5 · 4 20	10 · 16 160	10 · 256 2560	5 · 65536 327680	1 · 4294967296 4294967296	4295297718

PascalDrop with columns multiplied by Grass

💧 triangle CedarDrop row sums CrocusDrop
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	0	1 · 4 4					4
2	0	1 · 4 4	1 · 16 16				20
3	0	1 · 4 4	2 · 16 32	1 · 256 256			292
4	0	1 · 4 4	3 · 16 48	3 · 256 768	1 · 65536 65536		66356
5	0	1 · 4 4	4 · 16 64	6 · 256 1536	4 · 65536 262144	1 · 4294967296 4294967296	4295231044

half

Pascal with columns multiplied by HalfGrass

🌊 triangle HalfCedar row sums HalfCrocus (A246418)
k n	0	1	2	3	4	5	sums
0	1 · 1 1						1
1	1 · 1 1	1 · 2 2					3
2	1 · 1 1	2 · 2 4	1 · 8 8				13
3	1 · 1 1	3 · 2 6	3 · 8 24	1 · 128 128			159
4	1 · 1 1	4 · 2 8	6 · 8 48	4 · 128 512	1 · 32768 32768		33337
5	1 · 1 1	5 · 2 10	10 · 8 80	10 · 128 1280	5 · 32768 163840	1 · 2147483648 2147483648	2147648859

PascalDrop with columns multiplied by HalfGrass

💧 triangle HalfCedarDrop row sums HalfCrocusDrop
k n	0	1	2	3	4	5	sums
0	1 · 1 1						1
1	0	1 · 2 2					2
2	0	1 · 2 2	1 · 8 8				10
3	0	1 · 2 2	2 · 8 16	1 · 128 128			146
4	0	1 · 2 2	3 · 8 24	3 · 128 384	1 · 32768 32768		33178
5	0	1 · 2 2	4 · 8 32	6 · 128 768	4 · 32768 131072	1 · 2147483648 2147483648	2147615522

illustration of HalfCedarDrop with male Boolean functions

See Gender of Boolean functions.

These matrices show the 97 females and 159 males among the first 256 Boolean functions — above as Zhegalkin indices, below as truth tables.
The male truth tables on the left side of the matrix contain repetitions, i.e. their pattern can be described by a shorter truth table.

These images show the blue patterns also seen in the two 8×256 matrices. (Only a few of the 128 on the right side of the matrix are shown.)
They are ordered by adicity (vertical) and valency (horizontal).

(a, v) ↦ ♂

HalfCedar(a, v)
HalfCedarDrop(a, v)

is the number of male BF with

arity
adicity

a and valency v.

Triangle SignedCedar

signed triangles
	base triangle	families			BF
	base triangle	diagonal	triangle	total	diagonal	triangle	total
strong	Turned Rhodo.	Thyme	SignedBerberis	Aloe	Grass	SignedBramble	ExAloe
balanced & strong	Turned Rhodo.	Saffron	SignedArdisia	Agave	Sesame	SignedAronia	ExAgave
dense	Pascal	Thyme	SignedCypress	Lily	Grass	SignedCedar	Lotus
balanced & dense	Pascal	Saffron	SignedOlearia	Chamomile	Sesame	SignedOleander	Calendula

The absolute values are Pascal with columns multiplied by Grass.

triangle SignedCedar row sums Lotus (A000371)
k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	1 · 2 −2	1 · 4 4					2
2	1 · 2 2	2 · 4 −8	1 · 16 16				10
3	1 · 2 −2	3 · 4 12	3 · 16 −48	1 · 256 256			218
4	1 · 2 2	4 · 4 −16	6 · 16 96	4 · 256 −1024	1 · 65536 65536		64594
5	1 · 2 −2	5 · 4 20	10 · 16 −160	10 · 256 2560	5 · 65536 −327680	1 · 4294967296 4294967296	4294642034

half

The absolute values are Pascal with columns multiplied by HalfGrass.

triangle HalfSignedCedar row sums HalfLotus (A003465)
k n	0	1	2	3	4	5	sums
0	1 · 1 1						1
1	1 · 1 −1	1 · 2 2					1
2	1 · 1 1	2 · 2 −4	1 · 8 8				5
3	1 · 1 −1	3 · 2 6	3 · 8 −24	1 · 128 128			109
4	1 · 1 1	4 · 2 −8	6 · 8 48	4 · 128 −512	1 · 32768 32768		32297
5	1 · 1 −1	5 · 2 10	10 · 8 −80	10 · 128 1280	5 · 32768 −163840	1 · 2147483648 2147483648	2147321017

Triangles Cedar(Drop)

Triangle SignedCedar

Triangles Cedar_(Drop)