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All-1's sequence
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(Redirected from Zeroth powers)
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, ...}
Contents
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
where is the zeta function, generates (enumerates) the all-1's sequence.
Notes
- ↑ Sondow, Jonathan and Weisstein, Eric W., Riemann Zeta Function, From MathWorld — A Wolfram Web Resource.