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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 gives the all-1's sequence.

Powers of one

The sequence of powers of one gives the all-1's sequence.

Simple continued fraction for golden ratio

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

has the all-1's sequence for the integer part and partial quotients

Generating functions

Ordinary generating function

The ordinary generating function

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

Exponential generating function

The exponential generating function

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

Dirichlet generating function

The Dirichlet generating function

[1]

where 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.