nvec ← num ##.to num                        ⍝ Sequence ⍺ .. ⍵

Returns a vector of numbers ⍺ to ⍵ inclusive. ⍺ is the _first_ value in the seq-
uence, and ⍵ the _last_. More generally, ⍺ may be a 2-item vector, which determ-
ines the first two items of the sequence and hence the _step_ size.

Notice  that  any of _first_, _step_ and _last_ may be positive, negative and/or
fractional.  If  _last_  is  not  a  whole  number  of _step_s from _first_, the
sequence will stop short of _last_.

Technical notes:

The following simple function suffices for the ascending sequence ⍺, ⍺+1, ··· ⍵.
Notice how (·-⎕io) makes it independent of index origin:

    to←{(⍺-⎕io)+⍳1+⍵-⍺}

Useful  refinements  to  the  function allow _descending_ sequences and optional
_step_ sizes other than 1:

    to←{⎕io←0                                   ⍝ Sequence ⍺ .. ⍵
        from step←1 ¯1×-\2↑⍺,⍺+×⍵-⍺             ⍝ step default is +/- 1.
        from+step×⍳0⌈1+⌊(⍵-from)÷step+step=0    ⍝ ⍺ thru ⍵ inclusive.
    }

Attempts to re-cast the function as a one-liner, include:

    to←{From+0,+\Step⍴⍨0⌈⌊Inc÷Step←--/2⍴⍺,From+×Inc←⍵-From←1⍴⍺}

    to←{↑+/⍵{⍵×{⍵-⎕IO}⍳1+0⌈⌊(⍺⍺-⍺)÷⍵+⍵=0}\1 ¯1×-\2↑⍺,⍺+×⍵-⍺}

    to←{(⍬⍴⍺)+(|-/⌽2↑⍺,⍺+1){⍺×(×⍵)×(-⎕IO)+⍳1+⌊|⍵÷⍺}⍵-⍬⍴⍺}

VMJ  suggests the following extension, which accepts an optional explicit _step_
size  as  a second item of its _right_ argument. _step_ defers to _next_ if both
are supplied.

    to←{
        ⎕IO←0 ⋄ ⍺←1 ⋄ z←0.0000000001
        (a n)←2↑⍺,⍺ ⋄ (b s)←2↑⍵,1
        d←|s{⍵≠0:⍵ ⋄ ⍺≠0:⍺ ⋄ 1}n-a
        {⍵×z<|0-⍵}a+d{(⍺××⍵)×⍳1+⌊|⍵÷⍺}b-a
    }

    1 3 to 5      ⍝ using: _next_
1 3 5

    1 to 5 2      ⍝ using: _step_
1 3 5

Bob Smith suggests this extension for nested sequences:

    to←{                                    ⍝ Sequence ⍺ .. ⍵
        from step←⊂¨1 ¯1×-\2↑⍺,⍺+×⍵-⍺       ⍝ step default is +/- 1.
        size←0⌈1+⌊⊃(⍵-from)÷step+step=0     ⍝ shape of result
        from+step×(⍳size)-⎕io               ⍝ ⍺ thru ⍵ inclusive.
    }

    disp (¯1 6)(0 4) to ⊂3 ¯2
┌→───┬────┬────┬────┬─────┐
↓¯1 6│¯1 4│¯1 2│¯1 0│¯1 ¯2│
├~──→┼~──→┼~──→┼~──→┼~───→┤
│0 6 │0 4 │0 2 │0 0 │0 ¯2 │
├~──→┼~──→┼~──→┼~──→┼~───→┤
│1 6 │1 4 │1 2 │1 0 │1 ¯2 │
├~──→┼~──→┼~──→┼~──→┼~───→┤
│2 6 │2 4 │2 2 │2 0 │2 ¯2 │
├~──→┼~──→┼~──→┼~──→┼~───→┤
│3 6 │3 4 │3 2 │3 0 │3 ¯2 │
└~──→┴~──→┴~──→┴~──→┴~───→┘

Examples:

      3 to 10               ⍝ Inclusive ascending sequence.
3 4 5 6 7 8 9 10

      10 to 3               ⍝ Descending sequence.
10 9 8 7 6 5 4 3

      7 to 7                ⍝ Single-item sequence.
7

      5 7 to 13             ⍝ 2-item start determines step.
5 7 9 11 13

      ¯10 ¯15 to ¯25        ⍝ Negative start and step.
¯10 ¯15 ¯20 ¯25

      1.5 1.7 to 2.5        ⍝ Fractional start and step.
1.5 1.7 1.9 2.1 2.3 2.5

      0 3 to 10             ⍝ Sequence stops short of _last_.
0 3 6 9

See also: sieve

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