Editing Figurate number (section)

== Triangular numbers and their analogs in higher dimensions ==

The [[triangular number]]s for {{math|''n'' {{=}} 1, 2, 3, ...}} are the result of the juxtaposition of the linear numbers (linear gnomons) for {{math|''n'' {{=}} 1, 2, 3, ...}}:
{| cellpadding="10"
|- align="center" valign="bottom"
|style="display: inline-block; line-height: 0;"| [[File:GrayDotX.svg|16px|*]]
|style="display: inline-block; line-height: 0;"| [[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
|style="display: inline-block; line-height: 0;"| [[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
|style="display: inline-block; line-height: 0;"| [[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
|style="display: inline-block; line-height: 0;"| [[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
|style="display: inline-block; line-height: 0;"| [[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]<br />[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
|}

These are the binomial coefficients <math>\textstyle \binom{n+1}{2} </math>. This is the case {{math|''r'' {{=}} 2}} of the fact that the {{mvar|r}}th diagonal of [[Pascal's triangle#Patterns and properties|Pascal's triangle]] for {{math|''r'' ≥ 0}} consists of the figurate numbers for the {{mvar|r}}-dimensional analogs of triangles ({{mvar|r}}-dimensional [[simplex|simplices]]).

The simplicial polytopic numbers for {{math|''r'' {{=}} 1, 2, 3, 4, ...}} are:
*<math>P_1(n) = \frac{n}{1} = \binom{n+0}{1}=\binom{n}{1} </math> (linear numbers),
*<math>P_2(n) = \frac{n(n+1)}{2} = \binom{n+1}{2}</math> ([[triangular number]]s),
*<math>P_3(n) = \frac{n(n+1)(n+2)}{6} = \binom{n+2}{3}</math> ([[tetrahedral number]]s),
*<math>P_4(n) = \frac{n(n+1)(n+2)(n+3)}{24} = \binom{n+3}{4}</math> (pentachoric numbers, [[pentatopic number]]s, 4-simplex numbers),
<math>\qquad\vdots</math>
*<math>P_r(n) = \frac{n(n+1)(n+2)\cdots(n+r-1)}{r!} = \binom{n+(r-1)}{r}</math> ({{mvar|r}}-topic numbers, {{mvar|r}}-[[simplex]] numbers).

The terms ''[[square number]]'' and ''[[cubic number]]'' derive from their geometric representation as a [[Square (geometry)|square]] or [[cube (geometry)|cube]]. The difference of two positive triangular numbers is a [[trapezoidal number]].