

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
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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



