Sequence Heather

Studies of Boolean functions sequences related to seals

Clover is the summation of Heather, which is the summation of Lavender. Compare triangle Lime.

Sequences Clover and Heather

	0	1	2	3	4	5	6	7
🌊 Clover (A076766)	1	2	4	8	16	32	68	148
💧 Heather (~A034343)	1	1	2	4	8	16	36	80

Compare MaimedClover.            The differences of Clover and the powers of two are A250003.

Heather appears in the following triangles: StackedHeather, MaimedStackedHeather DiagonalHeather, SlopedHeather, MaimedSlopedHeather

row sums of Elm and Lime
🌊    triangle Elm (A076831)    row sums Clover (A076766) k n 0 1 2 3 4 5 6 7 sums 0 1 1 1 1 1 2 2 1 2 1 4 3 1 3 3 1 8 4 1 4 6 4 1 16 5 1 5 10 10 5 1 32 6 1 6 16 22 16 6 1 68 7 1 7 23 43 43 23 7 1 148 💧    triangle Lime (~A034253)    row sums Heather (~A034343) k n 0 1 2 3 4 5 6 7 sums 0 1 1 1 0 1 1 2 0 1 1 2 3 0 1 2 1 4 4 0 1 3 3 1 8 5 0 1 4 6 4 1 16 6 0 1 6 12 11 5 1 36 7 0 1 7 21 27 17 6 1 80

row sums of StackedHeather and DiagonalHeather
🌊    triangle StackedHeather    row sums Clover k n 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 k n 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

row sums of SlopedHeather and SlopedLavender
🌊    triangle SlopedHeather    row sums Clover k n 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 k n 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 ↦ houses

Clover(a) Heather(a)

is the number of houses of the seals with

arity adicity

a.

Sequences Heather and Lavender

	0	1	2	3	4	5	6	7
🌊 Heather (~A034343)	1	1	2	4	8	16	36	80
💧 Lavender	1	0	1	2	4	8	20	44

row sums of TurnedLime and Magnolia

🌊    triangle TurnedLime (~A034253)     row sums Heather (~A034343) k n 0 1 2 3 4 5 6 7 8 9 10 11 12 sums 0 1 1 1 1 0 1 2 1 1 0 2 3 1 2 1 0 4 4 1 3 3 1 0 8 5 1 4 6 4 1 0 16 6 1 5 11 12 6 1 0 36 7 1 6 17 27 21 7 1 0 80 8 1 7 25 54 63 34 9 1 0 194 9 1 8 35 99 163 134 54 11 1 0 506 10 1 9 47 170 385 465 276 82 13 1 0 1449 11 1 10 61 277 847 1472 1283 544 120 15 1 0 4631 12 1 11 78 436 1775 4408 5676 3480 1048 174 18 1 0 17106   💧    triangle Magnolia    row sums Lavender k n 0 1 2 3 4 5 6 7 8 9 10 11 12 sums 0 1 1 1 0 0 0 2 0 1 0 1 3 0 1 1 0 2 4 0 1 2 1 0 4 5 0 1 3 3 1 0 8 6 0 1 5 8 5 1 0 20 7 0 1 6 15 15 6 1 0 44 8 0 1 8 27 42 27 8 1 0 114 9 0 1 10 45 100 100 45 10 1 0 312 10 0 1 12 71 222 331 222 71 12 1 0 943 11 0 1 14 107 462 1007 1007 462 107 14 1 0 3182 12 0 1 17 159 928 2936 4393 2936 928 159 17 1 0 12475

a−g ↦ houses

Heather(a−g) Lavender(a−g)

is the number of houses of the seals with

arity adicity

a and gravity g.   (See triangle pair SlopedHeather.)