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
Famous quotes containing the word combinations:
“The more elevated a culture, the richer its language. The number of words and their combinations depends directly on a sum of conceptions and ideas; without the latter there can be no understandings, no definitions, and, as a result, no reason to enrich a language.”
—Anton Pavlovich Chekhov (1860–1904)
“Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.”
—Paul Valéry (1871–1945)
“The wider the range of possibilities we offer children, the more intense will be their motivations and the richer their experiences. We must widen the range of topics and goals, the types of situations we offer and their degree of structure, the kinds and combinations of resources and materials, and the possible interactions with things, peers, and adults.”
—Loris Malaguzzi (1920–1994)