Triangular numbers: Difference between revisions
Content added Content deleted
(Added Wren) |
Thundergnat (talk | contribs) m (minor clarifications) |
||
Line 15: | Line 15: | ||
root are triangular numbers. |
root are triangular numbers. |
||
The real triangular root of a number may be found using: <math>n = \frac{\sqrt{8x+1}-1}{2}</math> |
The real triangular root of a number ''x'' may be found using: <math>n = \frac{\sqrt{8x+1}-1}{2}</math> |
||
Line 26: | Line 26: | ||
Or, may be calculated directly: <math>Te_n= \sum_{k=1}^n T_k =\frac{n(n+1)(n+2)}{6} = \frac{n^{\overline 3}}{3!} = \binom{n+2}{3}</math> (Binomial "n plus two choose three".) |
Or, may be calculated directly: <math>Te_n= \sum_{k=1}^n T_k =\frac{n(n+1)(n+2)}{6} = \frac{n^{\overline 3}}{3!} = \binom{n+2}{3}</math> (Binomial "n plus two choose three".) |
||
One may find the real tetrahedral root using the formula: <math>n = \sqrt[3]{3x+\sqrt{9{x^2}-\frac{1}{27}}} +\sqrt[3]{3x-\sqrt{9{x^2}-\frac{1}{27}}} -1</math> |
One may find the real tetrahedral root of ''x'' using the formula: <math>n = \sqrt[3]{3x+\sqrt{9{x^2}-\frac{1}{27}}} +\sqrt[3]{3x-\sqrt{9{x^2}-\frac{1}{27}}} -1</math> |
||
<br>''Depending on the math precision of your particular language, may need to be rounded to the nearest 1e-16 or so.'' |
<br>''Depending on the math precision of your particular language, may need to be rounded to the nearest 1e-16 or so.'' |
||
Line 35: | Line 35: | ||
Again, the ''nth'' pentatope is the sum of the first '''n''' tetrahedral numbers, or <math>P_n = \frac{n(n+1)(n+2)(n+3)}{24} = \binom{n + 3}{4}</math> (Binomial "n plus three choose four".) |
Again, the ''nth'' pentatope is the sum of the first '''n''' tetrahedral numbers, or <math>P_n = \frac{n(n+1)(n+2)(n+3)}{24} = \binom{n + 3}{4}</math> (Binomial "n plus three choose four".) |
||
The pentatopic real root of ''x'' may be found using: <math>n = \frac{\sqrt{5+4\sqrt{24x+1}} - 3}{2}.</math> |
|||
In general, these all belong to the class [[wp:Figurate_number|figurate numbers]] as they are |
In general, these all belong to the class [[wp:Figurate_number|figurate numbers]] as they are |
||
based on '''r''' dimensional figures. Sometimes they are referred to as '''r-simplex''' |
based on '''r''' dimensional geometric figures. Sometimes they are referred to as '''r-simplex''' |
||
numbers. In geometry a [[wp:Simplex|simplex]] is the simplest '''r-dimensional''' |
numbers. In geometry a [[wp:Simplex|simplex]] is the simplest '''r-dimensional''' |
||
object possible. |
object possible. |
||
Line 64: | Line 64: | ||
;* [[wp:Triangular_number|Wikipedia: Triangular numbers]] |
;* [[wp:Triangular_number|Wikipedia: Triangular numbers]] |
||
;* [[wp:Tetrahedral_number|Wikipedia: Tetrahedral numbers]] |
;* [[wp:Tetrahedral_number|Wikipedia: Tetrahedral numbers]] |
||
;* [[wp:Pentatope_number|Wikipedia: |
;* [[wp:Pentatope_number|Wikipedia: Pentatopic numbers]] |
||
;* [[wp:Figurate_number|Wikipedia: Figurate numbers]] |
;* [[wp:Figurate_number|Wikipedia: Figurate numbers]] |
||
;* [[wp:Simplex|Wikipedia: Simplex(geometry)]] |
;* [[wp:Simplex|Wikipedia: Simplex(geometry)]] |