E -> Ea Eb
Ea -> Function
      U_op E
      Property
Eb -> P_op E
      whitespace

E -> Ea Eb
     E ? E : E
Ea -> Function
      U_op E
      Property
Eb -> P_op E
      whitespace

E -> Ea Eb
     EaEb ? E : E
Ea -> Function
      U_op E
      Property
Eb -> P_op E
      whitespace

# pp. 1517-1751: GAP-generated tables.

f(1) = [false]
f(2) = [true]
f(3) = [false, false]
f(4) = [true, false]
f(5) = [true, true]
f(6) = [false, false, false]
f(7) = [true, false, false]
f(8) = [true, true, false]
f(9) = [true, true, true]
f(10) = [false, false, false, false]

declare @ int=32;while(@<=126)begin;print char(@);set @+=1;end

boolean[] f(int val) {
	int length = 0;
	for (; length < val; length++, val -= length);

	boolean[] set = new boolean[length];

	for (int i = 0; val > 0; i++, val--) {
		set[i] = true;
	}

	return set;
}

for (; length < val; length++) {
    val -= length;
}

// README! This is needed because ...

var totalsize = ((Math.sqrt(1+8*n)-1)/2)|0;
var noftrues  = ((Math.sqrt(8*n-7)-1)/2)|0;
Array(noftrues).fill(true).concat(Array(totalsize-noftrues).fill(false))

0,
1, 1,
2, 2, 2,
3, 3, 3, 3,
4, 4, 4, 4, 4,
5, 5, 5, 5, 5, 5,
6, 6, 6, 6, 6, 6, 6,
7, 7, 7, 7, 7, 7, 7, 7,

floor((sqrt(1+8*n)-1)/2)

0;
0, 1;
0, 1, 2;
0, 1, 2, 3;
0, 1, 2, 3, 4;
0, 1, 2, 3, 4, 5;
0, 1, 2, 3, 4, 5, 6;
0, 1, 2, 3, 4, 5, 6, 7;
0, 1, 2, 3, 4, 5, 6, 7, 8;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;

Floor[(-1 + Sqrt[8 n - 7])/2]

f(1) = [true]
f(2) = [true]
f(3) = [true, true]
f(4) = [true, true]
f(5) = [true, true]
f(6) = [true, true, true]
f(7) = [true, true, true]
f(8) = [true, true, true]
f(9) = [true, true, true]
f(10) = [true, true, true, true]

function f(n) {
  var totalsize = Math.floor((Math.sqrt(1+8*n)-1)/2);
  n++;
  var _nft = Math.floor((-1 + Math.sqrt(8*n - 7))/2);
  var noftrues = (n - _nft * (_nft + 1)/2) - 1;
  return Array(noftrues).fill(true).concat(Array(totalsize - noftrues).fill(false))
}

function f(n) {
	var totalsize = Math.floor((Math.sqrt(1+8*n)-1)/2);
	var noftrues = (n - totalsize * (totalsize + 1)/2);
	return Array(noftrues).fill(true).concat(Array(totalsize - noftrues).fill(false))
}

f(1) = [false]
f(2) = [false]
f(3) = [true, false]
f(4) = [false, false]
f(5) = [true, false]
f(6) = [true, true, false]
f(7) = [false, false, false]
f(8) = [true, false, false]
f(9) = [true, true, false]
f(10) = [true, true, true, false]

> function f(n) {
var totalsize = Math.floor((Math.sqrt(1+8*n)-1)/2);
var _nft = Math.floor((Math.sqrt(8*n+1)-1)/2);
var noftrues = (n - _nft * (_nft + 1)/2);
return Array(noftrues).fill(true).concat(Array(totalsize - noftrues).fill(false))
}
for (var i = 1; i < 11; i++) {
console.log(f(i));
}
[Log] [false] (1)
[Log] [true] (1)
[Log] [false, false] (2)
[Log] [true, false] (2)
[Log] [true, true] (2)
[Log] [false, false, false] (3)
[Log] [true, false, false] (3)
[Log] [true, true, false] (3)
[Log] [true, true, true] (3)

f(1) = [false]
f(2) = [true]
f(3) = [false, false]
f(4) = [true, false]
f(5) = [true, true]
f(6) = [false, false, false]
f(7) = [true, false, false]
f(8) = [true, true, false]
f(9) = [true, true, true]
f(10) = [false, false, false, false]

php > foreach(range(32,128)as $i)echo chr($i);
 !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~?

-> (foo, bar) { baz }

T -> Ea Eb ? E : E
E -> T
     Ea Eb

// TODO write readable code