User:Charles R Greathouse IV/Standards

These are my personal editorial standards for various types of sequences. If you are submitting one of these types of sequences, I recommend following these guidelines when possible.

Index

Use it! See User:Charles R Greathouse IV/Metadata#Index for some of my thoughts.

References and links

References and links should be referred to in the text (mainly in comments and formulas) by all authors' names and a disambiguator like year or first name only as needed. I follow Guy's convention of using ampersand (&) between authors of a single paper or other work and the spelled-out word "and" when referring to multiple works. Thus

a(n) ~ n, see Ant & Bee link.

but

This sequence is complete, see Camel (1999) and Dove links.

if there is a paper by Ant and Bee together, two or more papers by Camel (only one of which published in 1999), and a separate paper of Dove.

Asymptotics

I make heavy use of Big O notation, Vinogradov notation, asymptotic equivalence ~, and occasional use of Little-o notation and ≍. I try to avoid big-Omega (Ω), little-Omega (ω), big-Theta (ϴ), and soft-O notation (Õ).

Vinogradov notation

Generally, if f(x) = O(g(x)), we can write $f(x)\ll g(x)$ and $g(x)\gg f(x)$. In the OEIS these should be written as << and >> rather than with the Unicode ≪ and ≫. The LaTeX equivalents are \ll and \gg.

Equivalence

Two functions f(x) and g(x) are strictly asymptotically equivalent if lim f(x)/g(x) = 1. This can be written f(x) ~ g(x). The LaTeX equivalent is \sim.

A weaker notion is f(x) ≍ g(x) which means that the lim inf of f(x)/g(x) is positive and the lim sup is finite. The LaTeX equivalent is \asymp. This can also be written f(x) = Theta(g(x)) or f(x) = ϴ(g(x)) but this can be confused for a large number of other functions (Chebyshev theta or Jacobi theta being perhaps the most popular) so I avoid it.

Note that f(x) ≍ g(x) is equivalent to g(x) << f(x) << g(x), so it's possible to avoid its use. It can even be written 1 << f(x)/g(x) << 1 if g(x) is nonzero for large enough x (as is typically the case in applications). The particular choice should depend on which way of writing it is likely to be most clear to readers.

Constants

In most cases, the primary sequence for a constant should be its decimal expansion. The remarks below apply to this case; other associated sequences (continued fraction representation, binary expansion, Engel expansion, etc.) should be mentioned in the crossreferences. Such constants should have the keyword cons and have terms exclusively from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} (some, of course, will not use all of these). Further, such sequences should never have the keyword sign (see Keyword compatibility).

Rational

Give the number's numerator and denominator unless they are large; otherwise, give the number of digits in each if possible. Give the period of the decimal expansion unless terminating (in which case so state).

If the number is an integer, instead say so (unless this fact is very much obvious from the definition) and give its length in decimal digits (approximately, if necessary). All such numbers should have keywords cons and fini (though many such are merely terminating rationals).

Algebraic Irrational

Please link in all cases to the Index:

<a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

adjusting the degree as appropriate. Consider giving the number's minimal polynomial (if sufficiently small) and its denominator (the smallest positive integer k such that kx is an algebraic integer). If the denominator is 1, instead write that it is an algebraic integer.

Transcendental

It can be useful to note that a number is transcendental, either by a standard theorem (please give the theorem for future readers!) or by a reference/link. This is not necessary for very obvious transcendental numbers, e.g., rational multiples of Pi, but please link in all cases to the Index:

<a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

Unknown

If the status of the number has been explicitly mentioned as open/unknown or conjectured in a paper, please mention this in the sequence and cite the paper.