# Triangle Lime

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) :
| kn | 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) :
| kn | 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 |
Elm(n, k) is the number of houses with arity n and depth k. Lime(n, k) is the number of houses with adicity n and depth k.
The rows of Lime sum up to those of Tilia.
columns of Elm :
| 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) :
| kn | 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 :
| kn | 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 |
All entries in WisteriaDrop come from triangle Magnolia. In Wisteria that is almost the same, but some entries are multiplied.
Magnolia as a square array :
SquareMagnolia :
| kn | 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).