# Triangle Poplar

| | valency gravity | strength depth |
| triangle | Willow | Poplar |
| diagonal | Lotus | Lily |
| difference | Orchid | Oxalis |
* Triangle Oak with multiplied columns

## Triangles Poplar$_{(Drop)}$

The diagonal of both triangles is Lily.
🌊    triangle Poplar    row sums Grass :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 1 · 2 2 | 1 · 2 2 | | | | | 4 |
| 2 | 1 · 2 2 | 3 · 2 6 | 1 · 8 8 | | | | 16 |
| 3 | 1 · 2 2 | 7 · 2 14 | 7 · 8 56 | 1 · 184 184 | | | 256 |
| 4 | 1 · 2 2 | 15 · 2 30 | 35 · 8 280 | 15 · 184 2760 | 1 · 62464 62464 | | 65536 |
| 5 | 1 · 2 2 | 31 · 2 62 | 155 · 8 1240 | 155 · 184 28520 | 31 · 62464 1936384 | 1 · 4293001088 4293001088 | 4294967296 |
💧    triangle PoplarDrop    row sums GrassDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 2 · 2 4 | 1 · 8 8 | | | | 12 |
| 3 | 0 | 4 · 2 8 | 6 · 8 48 | 1 · 184 184 | | | 240 |
| 4 | 0 | 8 · 2 16 | 28 · 8 224 | 14 · 184 2576 | 1 · 62464 62464 | | 65280 |
| 5 | 0 | 16 · 2 32 | 120 · 8 960 | 140 · 184 25760 | 30 · 62464 1873920 | 1 · 4293001088 4293001088 | 4294901760 |
💧    triangle PoplarDrop    row sums GrassDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 2 · 1 2 | | | | | | 2 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 1 · 4 4 | 2 · 4 8 | | | | 12 |
| 3 | 0 | 1 · 8 8 | 6 · 8 48 | 23 · 8 184 | | | 240 |
| 4 | 0 | 1 · 16 16 | 14 · 16 224 | 161 · 16 2576 | 3904 · 16 62464 | | 65280 |
| 5 | 0 | 1 · 32 32 | 30 · 32 960 | 805 · 32 25760 | 58560 · 32 1873920 | 134156284 · 32 4293001088 | 4294901760 |
half :
🌊    triangle HalfPoplar    row sums HalfGrass :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 1 1 | | | | | 2 |
| 2 | 1 · 1 1 | 3 · 1 3 | 1 · 4 4 | | | | 8 |
| 3 | 1 · 1 1 | 7 · 1 7 | 7 · 4 28 | 1 · 92 92 | | | 128 |
| 4 | 1 · 1 1 | 15 · 1 15 | 35 · 4 140 | 15 · 92 1380 | 1 · 31232 31232 | | 32768 |
| 5 | 1 · 1 1 | 31 · 1 31 | 155 · 4 620 | 155 · 92 14260 | 31 · 31232 968192 | 1 · 2146500544 2146500544 | 2147483648 |
💧    triangle HalfPoplarDrop    row sums HalfGrassDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 0 | 1 · 1 1 | | | | | 1 |
| 2 | 0 | 2 · 1 2 | 1 · 4 4 | | | | 6 |
| 3 | 0 | 4 · 1 4 | 6 · 4 24 | 1 · 92 92 | | | 120 |
| 4 | 0 | 8 · 1 8 | 28 · 4 112 | 14 · 92 1288 | 1 · 31232 31232 | | 32640 |
| 5 | 0 | 16 · 1 16 | 120 · 4 480 | 140 · 92 12880 | 30 · 31232 936960 | 1 · 2146500544 2146500544 | 2147450880 |
The columns of Poplar$_{(Drop)}$ are those of Olive$_{(Drop)}$ multiplied by powers of two.

## Triangles TurnedPoplar$_{(Drop)}$

Column 0 is Lily$_{(Drop)}$.  The row sums without it are Oxalis$_{(Drop)}$.
🌊    triangle TurnedPoplar    row sums Grass :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 2 | | | | | | 2 |
| 1 | 2 | 2 | | | | | 4 |
| 2 | 8 | 6 | 2 | | | | 16 |
| 3 | 184 | 56 | 14 | 2 | | | 256 |
| 4 | 62464 | 2760 | 280 | 30 | 2 | | 65536 |
| 5 | 4293001088 | 1936384 | 28520 | 1240 | 62 | 2 | 4294967296 |
💧    triangle TurnedPoplarDrop    row sums GrassDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 2 | | | | | | 2 |
| 1 | 0 | 2 | | | | | 2 |
| 2 | 6 | 4 | 2 | | | | 12 |
| 3 | 176 | 50 | 12 | 2 | | | 240 |
| 4 | 62280 | 2704 | 266 | 28 | 2 | | 65280 |
| 5 | 4292938624 | 1933624 | 28240 | 1210 | 60 | 2 | 4294901760 |

## Triangles SmallTurnedPoplar$_{(Drop)}$

Column 0 is SmallLily$_{(Drop)}$.  The row sums without it are SmallOxalis$_{(Drop)}$.
🌊    triangle SmallTurnedPoplar    row sums SmallGrass :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 | | | | | | 1 |
| 1 | 2 | 1 | | | | | 3 |
| 2 | 8 | 6 | 1 | | | | 15 |
| 3 | 184 | 56 | 14 | 1 | | | 255 |
| 4 | 62464 | 2760 | 280 | 30 | 1 | | 65535 |
| 5 | 4293001088 | 1936384 | 28520 | 1240 | 62 | 1 | 4294967295 |
💧    triangle SmallTurnedPoplarDrop    row sums SmallGrassDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 | | | | | | 1 |
| 1 | 1 | 1 | | | | | 2 |
| 2 | 6 | 5 | 1 | | | | 12 |
| 3 | 176 | 50 | 13 | 1 | | | 240 |
| 4 | 62280 | 2704 | 266 | 29 | 1 | | 65280 |
| 5 | 4292938624 | 1933624 | 28240 | 1210 | 61 | 1 | 4294901760 |