Triangle Lime

Studies of Boolean functions
sequences related to seals

Elm is the summation of Lime, which is the turned summation of Magnolia. Compare sequence Heather.

Triangles with n up to 25 are shown on this subpage.

Triangles Elm and Lime

Elm should not be confused with Pascal's triangle. Its contains bigger entries in rows ≥ 6. The differences are in A250002.

🌊 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

Compare triangle MaimedElm.

The rows of Lime sum up to those of Tilia.

(a, d) ↦ houses

Elm(a, d)
Lime(a, d)

is the number of houses of the seals with

arity
adicity

a and depth d.

columns of Elm

The OEIS shows a formula only for column 2, namely $\left\lfloor {\frac {2\cdot (2n^{2}+21n-6)}{72}}\right\rfloor$ .
Generating functions are shown or linked for columns up to 6 (which is a whole page long).

0		1, 1, 1...
1		1, 2, 3...
2	A034198	1, 3, 6, 10, 16, 23, 32, 43, 56, 71, 89, 109, 132, 158, 187, 219, 255...
3	A034357	1, 4, 10, 22, 43, 77, 131, 213, 333, 507, 751, 1088, 1546, 2159, 2967, 4023, 5384, 7122...
4	A034358	1, 5, 16, 43, 106, 240, 516, 1060, 2108, 4064, 7641, 14036, 25253, 44560, 77245, 131658...
5	A034359	1, 6, 23, 77, 240, 705, 1988, 5468, 14724, 39006, 101818, 261924, 663748, 1655781, 4062110...
6	A034360	1, 7, 32, 131, 516, 1988, 7664, 29765, 117169, 467266, 1880517, 7588675, 30491836...
7	A034361	1, 8, 43, 213, 1060, 5468, 29765, 173035, 1074526, 7059804, 48235007, 336048291...
8	A034362	1, 9, 56, 333, 2108, 14724, 117169, 1074526, 11249092, 130484439, 1612782351...

Triangles 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

pyramids in hyperpyramids Wisteria and WisteriaDrop

All entries in WisteriaDrop come from triangle Magnolia.
In Wisteria that is almost the same, but some entries are multiplied.

?	Do these triangles on their own count anything?

Magnolia as a square array

SquareMagnolia
k n	0	1	2	3	4	5	6	7	8	9	10	11	12
0	1	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	1	1	1	1	1	1	1	1	1	1
2	0	1	2	3	5	6	8	10	12	14	17
3	0	1	3	8	15	27	45	71	107	159
4	0	1	5	15	42	100	222	462	928
5	0	1	6	27	100	331	1007	2936
6	0	1	8	45	222	1007	4393
7	0	1	10	71	462	2936
8	0	1	12	107	928
9	0	1	14	159
10	0	1	17
11	0	1
12	0

The following two lines show some similarity between column 2 of Magnolia (above) and A045919, the partial sum of the Goldbach numbers (below).

0, 1, 2, 3, 5, 6, 8, 10, 12, 14, 17, 19, 22, 25, 28, 31, 35, 38,   42, 46,   50, 54, 59, 63
0, 1, 2, 3, 5, 6, 8, 10, 12, 14, 17, 20, 23, 25, 28, 30, 34, 38, 40, 43, 47, 50, 54, 59, 63