Triangular numbers: Difference between revisions
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minor clarifications
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root are triangular numbers.
The real triangular root of a number ''x'' may be found using: <math>n = \frac{\sqrt{8x+1}-1}{2}</math>
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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 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.''
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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".)
In general, these all belong to the class [[wp:Figurate_number|figurate numbers]] as they are
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'''
object possible.
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;* [[wp:Triangular_number|Wikipedia: Triangular numbers]]
;* [[wp:Tetrahedral_number|Wikipedia: Tetrahedral numbers]]
;* [[wp:Pentatope_number|Wikipedia:
;* [[wp:Figurate_number|Wikipedia: Figurate numbers]]
;* [[wp:Simplex|Wikipedia: Simplex(geometry)]]
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