## Blog Posts

If the Sum of the Digits of a Number is Divisible by Three, the Number Is Divisible by Three [August 22, 2019]

## Miscellaneous

Phi (\(\phi\)) is the golden ratio.

\({\Large \phi = \frac{ 1 + \sqrt{5} }{2}}\)

Look into Toads and Frogs game. See OEIS “A005563 a(n) = n*(n+2) = (n+1)^2 – 1″ or OEIS search for “frog”.

x^2+(5y/4-sqrt(abs(x)))^2=1

## Numbers

\(3^3 + 4^4 + 3^3 + 5^5 = 3435\) — The only other number with this property is 1.

## Notation

### \({}^n C {}_r\) — N Choose R

\({}^n C {}_r\) (“n choose r”) is an alternate notation for the binomial symbol:

$${}^n C {}_r = {n \choose r} = \frac{n!}{r!(n−r)!}$$

\({}^n C _r\) is an alternate notation for the binomial symbol \({n \choose r} = \frac{n!}{r!(n−r)!}\)

Inline style: `\({}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120\)`

\({}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120\)

Display style: `$${}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120$$`

$${}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120$$

### \(\sum_{n=1}^{\infty}\) — Summation

Inline style: `\(\sum_{n=1}^{\infty} 2^{-n} = 1\)`

\(\sum_{n=1}^{\infty} 2^{-n} = 1\)

Display style: `$$\sum_{n=1}^{\infty} 2^{-n} = 1$$`

$$\sum_{n=1}^{\infty} 2^{-n} = 1$$

Inline style: `\(\sum\limits_{k=1}^{n} k\)`

\(\sum\limits_{k=1}^{n} k\)

Display style: `$$\sum\limits_{k=1}^{n} k$$`

$$\sum\limits_{k=1}^{n} k$$

## See Also

Xcas, the swiss knife for mathematics. Giac/Xcas is a free computer algebra system for Windows, Mac OS X and Linux/Unix.

Use Xcas online in your web browser.

## References

Alekseyev, Max. “PARI/GP Scripts for Miscellaneous Math Problems“. *George Washington University*.

De Graeve, Renée; Parisse, Bernard. “Symbolic Algebra and Mathematics with Xcas“. *University of Grenoble I*.

“MathJax TeX and LaTeX Support“. *MathJax Consortium*.

Wikipedia contributors. “List of computer algebra systems“. *Wikipedia*.