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# All-1's sequence

A000012 The simplest sequence of positive integers: the all-1's sequence.

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}

## Zeroth powers

The sequence of zeroth powers ${\displaystyle \scriptstyle n^{0},\,n\,\geq \,0,\,}$ gives the all-1's sequence.

## Powers of one

The sequence of powers of one ${\displaystyle \scriptstyle 1^{n},\,n\,\geq \,0,\,}$ gives the all-1's sequence.

## Simple continued fraction for golden ratio

The simple continued fraction for the golden ratio (phi, A001622)

${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1+{\underset {i=1}{\overset {\infty }{\rm {K}}}}~{\frac {1}{1}}=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{a_{4}+\ddots }}}}}}}}\,}$

has the all-1's sequence for the integer part and partial quotients ${\displaystyle \scriptstyle \{a_{n}\}_{n=0}^{\infty }.\,}$

## Generating functions

### Ordinary generating function

${\displaystyle G_{\{n^{0}\}}(x)=G_{\{1^{n}\}}(x)=\sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}\,}$

generates (enumerates) the all-1's sequence.

### Exponential generating function

${\displaystyle E_{\{n^{0}\}}(x)=E_{\{1^{n}\}}(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=e^{x}\,}$

generates (enumerates) the all-1's sequence.

### Dirichlet generating function

${\displaystyle D_{\{n^{0}\}}(s)=D_{\{1^{n}\}}(s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\zeta (s)\,}$ [1]

where ${\displaystyle \scriptstyle \zeta (s)\,}$ is the zeta function, generates (enumerates) the all-1's sequence.

## Notes

1. Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld — A Wolfram Web Resource.