All-1's sequence

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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 $\scriptstyle n^{0},\,n\,\geq \,0,\,$ gives the all-1's sequence.

Powers of one

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

Simple continued fraction for golden ratio

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

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

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

Generating functions

Ordinary generating function

The ordinary generating function

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

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

Exponential generating function

The exponential generating function

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

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

Dirichlet generating function

The Dirichlet generating function

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

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