Editing Polygonal number (section)

==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 number]]s.

The following table summarizes the set of {{mvar|s}}-gonal {{mvar|t}}-gonal numbers for small values of {{mvar|s}} and {{mvar|t}}.
:{| class="wikitable" border="1"
|-
! {{mvar|s}}
! {{mvar|t}}
! Sequence
! [[On-Line Encyclopedia of Integer Sequences|OEIS]] number
|-
|4
|3
|1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600, 2507180834294496361, 85170343853180456676, 2893284510173841030625, 98286503002057414584576, 3338847817559778254844961, ...
| {{OEIS link|id=A001110}}
|-
|5
|3
|1, 210, 40755, 7906276, 1533776805, 297544793910, 57722156241751, 11197800766105800, 2172315626468283465, …
| {{OEIS link|id=A014979}}
|-
|5
|4
|1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, ...
| {{OEIS link|id=A036353}}
|-
|6
|3
|All hexagonal numbers are also triangular.
| {{OEIS link|id=A000384}}
|-
|6
|4
|1, 1225, 1413721, 1631432881, 1882672131025, 2172602007770041, 2507180834294496361, 2893284510173841030625, 3338847817559778254844961, 3853027488179473932250054441, ...
| {{OEIS link|id=A046177}}
|-
|6
|5
|1, 40755, 1533776805, …
| {{OEIS link|id=A046180}}
|-
|7
|3
|1, 55, 121771, 5720653, 12625478965, 593128762435, 1309034909945503, 61496776341083161, 135723357520344181225, 6376108764003055554511, 14072069153115290487843091, …
| {{OEIS link|id=A046194}}
|-
|7
|4
|1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609, 350709705290025, 25635978392186449, 9976444135331412025, …
| {{OEIS link|id=A036354}}
|-
|7
|5
|1, 4347, 16701685, 64167869935, …
| {{OEIS link|id=A048900}}
|-
|7
|6
|1, 121771, 12625478965, …
| {{OEIS link|id=A048903}}
|-
|8
|3
|1, 21, 11781, 203841, …
| {{OEIS link|id=A046183}}
|-
|8
|4
|1, 225, 43681, 8473921, 1643897025, 318907548961, 61866420601441, 12001766689130625, 2328280871270739841, 451674487259834398561, 87622522247536602581025, 16998317641534841066320321,  …
| {{OEIS link|id=A036428}}
|-
|8
|5
|1, 176, 1575425, 234631320, …
| {{OEIS link|id=A046189}}
|-
|8
|6
|1, 11781, 113123361, …
| {{OEIS link|id=A046192}}
|-
|8
|7
|1, 297045, 69010153345, …
| {{OEIS link|id=A048906}}
|-
|9
|3
|1, 325, 82621, 20985481, …
| {{OEIS link|id=A048909}} 
|-
|9
|4
|1, 9, 1089, 8281, 978121, 7436529, 878351769, 6677994961, 788758910641, 5996832038649, 708304623404049, 5385148492712041, 636056763057925561, ...
| {{OEIS link|id=A036411}} 
|-
|9
|5
|1, 651, 180868051, …
| {{OEIS link|id=A048915}} 
|-
|9
|6
|1, 325, 5330229625, …
| {{OEIS link|id=A048918}} 
|-
|9
|7
|1, 26884, 542041975, …
| {{OEIS link|id=A048921}} 
|-
|9
|8
|1, 631125, 286703855361, …
| {{OEIS link|id=A048924}} 
|-
|}

In some cases, such as {{math|''s'' {{=}} 10}} and {{math|''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 other such numbers has yet to be found.<ref>{{MathWorld|title=Pentagonal Square Triangular Number | urlname=PentagonalSquareTriangularNumber}}</ref>

The number [[1000_(number)#1200_to_1299|1225]] is hecatonicositetragonal ({{math|''s'' {{=}} 124}}), hexacontagonal ({{math|''s'' {{=}} 60}}), icosienneagonal ({{math|''s'' {{=}} 29}}), hexagonal, square, and triangular.