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PatternOperator

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Description

Summary

The pattern operator, ... (or …): between shown terms it names the
progression the shown terms determine, and the term after it names where the
progression stops. {1, 2, ..., n} is the integers from 1 to n,
{2, 4, ..., 2 n} the even ones, {5, 10, 15, ...} every positive multiple of
five, {..., -1, 0} the non-positive integers; 1 + 2 + ... + n is
sum(k, k, 1, n) and 1 * 2 * ... * n is product(k, k, 1, n).

Remarks

What is read, and what is refused. An arithmetic progression of whole numbers,
from at least two shown terms on one side of the dots: the step is the difference of the
first two, and every further shown term has to be on the progression, or the input is
refused with the term that is not. Three shown terms with no step in common,
{1, 4, 9, ...}, are a guess about the writer's rule and are refused rather than
answered with the smallest polynomial through them. A progression of anything but whole
numbers, {1/2, 1, 3/2, ..., n}, is refused too: a set of those is the image of a
range, which has no node yet.
What is built. Nothing new: a set is ZZ /\ [a; z] for step one and a residue
class cut by the interval, { x in ZZ : x = a (mod d) } /\ [a; z], otherwise -- both
list their members when the ends are numbers, and stay written over a symbolic end; a
one-sided pattern is the class cut by a ray. A sum is sum(a + d k, k, 0, (z - a)/d) and a product the same under product, with the index named k, or k_1 where the terms mention k. The general term with an index shown, a_1 + ... + a_n,
and the list of names of symbolic length, f(x_1, ..., x_n), are the larger half of
#1437 and wait for
v3's design of a family of names.

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