Combinations
Some numbers, such as 36 which is both square and triangular, fall into two polygonal sets. The problem of determining, given two such sets, all numbers that belong to both can be solved by reducing the problem to Pell's equation. The simplest example of this is the sequence of square triangular numbers.
The following table summarizes the set of s-gonal t-gonal numbers for small values of s and t.
| s | t | Sequence | OEIS number |
|---|---|---|---|
| 4 | 3 | 1, 36, 1225, 41616, … | A001110 |
| 5 | 3 | 1, 210, 40755, 7906276, … | A014979 |
| 5 | 4 | 1, 9801, 94109401, … | A036353 |
| 6 | 3 | All hexagonal numbers are also triangular. | A000384 |
| 6 | 4 | 1, 1225, 1413721, 1631432881, … | A046177 |
| 6 | 5 | 1, 40755, 1533776805, … | A046180 |
| 7 | 3 | 1, 55, 121771, 5720653, … | A046194 |
| 7 | 4 | 1, 81, 5929, 2307361, … | A036354 |
| 7 | 5 | 1, 4347, 16701685, 64167869935, … | A048900 |
| 7 | 6 | 1, 121771, 12625478965, … | A048903 |
| 8 | 3 | 1, 21, 11781, 203841, … | A046183 |
| 8 | 4 | 1, 225, 43681, 8473921, … | A036428 |
| 8 | 5 | 1, 176, 1575425, 234631320, … | A046189 |
| 8 | 6 | 1, 11781, 113123361, … | A046192 |
| 8 | 7 | 1, 297045, 69010153345, … | A048906 |
| 9 | 3 | 1, 325, 82621, 20985481, … | A048909 |
| 9 | 4 | 1, 9, 1089, 8281, 978121, … | A036411 |
| 9 | 5 | 1, 651, 180868051, … | A048915 |
| 9 | 6 | 1, 325, 5330229625, … | A048918 |
| 9 | 7 | 1, 26884, 542041975, … | A048921 |
| 9 | 8 | 1, 631125, 286703855361, … | A048924 |
In some cases, such as s=10 and t=4, there are no numbers in both sets other than 1.
The problem of finding numbers that belong to three polygonal sets is more difficult. A computer search for pentagonal square triangular numbers has yielded only the trivial value of 1, though a proof that there are no such number has yet to appear in print. All hexagonal square numbers are also hexagonal square triangular numbers, and 1225 is actually a hecticositetragonal, hexacontagonal, icosinonagonal, hexagonal, square, triangular number.
Read more about this topic: Polygonal Number
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