Triangular numbers
From specialfunctionswiki
The triangular numbers $T(n)$ are defined for $n=1,2,3,\ldots$ by the formula $$T(n)=\displaystyle\sum_{k=1}^n k.$$ They represent the number of ways to draw an equilateral triangle as in the image below.
Graphical representation of the first six triangular numbers.
Properties
T(n)=n(n+1)/2
T(n+1)=T(n)+n+1
n^2=T(n)+T(n-1)
T(n)^2=T(T(n))+T(T(n)-1)
T(n+1)^2-T(n)^2=(n+1)^3
References
- {{ #if: |{{{2}}}|V.E. Hoggatt, Jr}}{{#if: Marjorie Bicknell|{{#if: |, {{ #if: |{{{2}}}|Marjorie Bicknell}}{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and {{ #if: |{{{2}}}|Marjorie Bicknell}}}}|}}: [[Paper:V.E. Hoggatt, Jr/Triangular numbers{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|Triangular numbers{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}} (1974)| ({{#if: |{{{ed}}} ed., }}1974)}}]]{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: | ... [[{{{prev}}}|(previous)]]|}}{{#if: T(n)=n(n+1)/2 | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}} $(1.1)$
Triangular numbers
