A228650 Numbers k such that if an urn contains k balls, with at least one each of three colors, there exists a combination of the three colors such that it is equally probable for three balls randomly selected from the urn to all be either the same color or distinct colors.

6, 8, 11, 12, 46, 57, 66, 120, 121, 145, 156, 162, 166, 217, 372, 386, 557, 596, 638, 750, 866, 1025, 1038, 1201, 1396, 1857, 2042, 2081, 2146, 2263, 2301, 2452, 2836, 2900, 2926, 2991, 3026, 3053, 3288, 3368, 3963, 3970, 4511, 4656, 5006, 5492, 5890, 5952

Offset

1,1

Comments

If the urn contains 596 balls, there exist two inequivalent combinations with the desired property, {86, 246, 264} and {126, 154, 316}.

The analogous sequence for two colors are the square numbers > 1 (A000290 with first two terms truncated).

Links

Table of n, a(n) for n=1..48.

Example

46 is a member of the sequence because if the urn contains 6 red, 18 green and 22 blue balls, then there are 6 * 18 * 22 = 2376 selections of three balls with distinct colors, and ((6 * 5 * 4) + (18 * 17 * 16) + (22 * 21 * 20)) / 3! = 2376 selections of three balls all the same color, and 6 + 18 + 22 = 46.

Prog

(Pascal) program a228650;
var
  p: array[1..6000] of int64;
  b1, b2, b3, k: int64;
  n, s: integer;
begin
  k:=0;
  repeat
    inc(k);
    p[k] := (k * (k - 1) * (k - 2)) div 6;
  until k = 6000;
  n := 0; k := 2;
  repeat
    inc(k); s := 0;
    b1 := 0;
    repeat
      inc(b1);
      b2 := b1 - 1;
        b3 := k - (b1 + b2);
        repeat
        inc(b2); dec(b3);
        if (b3 >= b2) and (b1 * b2 * b3 = p[b1] + p[b2] + p[b3]) then
        begin
          inc(n); inc(s);
          writeln(n, ' ', k);
        end;
      until (b3 <= b2) or (s > 0);
    until (3 * b1 >= k) or (s > 0);
  until k = 6000;
end.

Crossrefs

Cf. A228651, A228652, A228653.

Sequence in context: A184113 A309627 A085412 * A358531 A200722 A120152

Adjacent sequences: A228647 A228648 A228649 * A228651 A228652 A228653

Keyword

nonn

Author

William Rex Marshall, Aug 29 2013

Status

approved