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A228650 Numbers n such that if an urn contains n 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. 4
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 (list; graph; refs; listen; history; text; internal format)
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 X 18 X 22 = 2376 selections of three balls with distinct colors, and ((6 X 5 X 4) + (18 X 17 X 16) + (22 X 21 X 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 * A200722 A120152 A315855

Adjacent sequences:  A228647 A228648 A228649 * A228651 A228652 A228653

KEYWORD

nonn

AUTHOR

William Rex Marshall, Aug 29 2013

STATUS

approved

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Last modified February 25 05:11 EST 2020. Contains 332217 sequences. (Running on oeis4.)