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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.

%I A228650 #17 Jul 30 2022 08:19:14
%S A228650 6,8,11,12,46,57,66,120,121,145,156,162,166,217,372,386,557,596,638,
%T A228650 750,866,1025,1038,1201,1396,1857,2042,2081,2146,2263,2301,2452,2836,
%U A228650 2900,2926,2991,3026,3053,3288,3368,3963,3970,4511,4656,5006,5492,5890,5952
%N 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.
%C A228650 If the urn contains 596 balls, there exist two inequivalent combinations with the desired property, {86, 246, 264} and {126, 154, 316}.
%C A228650 The analogous sequence for two colors are the square numbers > 1 (A000290 with first two terms truncated).
%e A228650 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.
%o A228650 (Pascal) program a228650;
%o A228650 var
%o A228650   p: array[1..6000] of int64;
%o A228650   b1, b2, b3, k: int64;
%o A228650   n, s: integer;
%o A228650 begin
%o A228650   k:=0;
%o A228650   repeat
%o A228650     inc(k);
%o A228650     p[k] := (k * (k - 1) * (k - 2)) div 6;
%o A228650   until k = 6000;
%o A228650   n := 0; k := 2;
%o A228650   repeat
%o A228650     inc(k); s := 0;
%o A228650     b1 := 0;
%o A228650     repeat
%o A228650       inc(b1);
%o A228650       b2 := b1 - 1;
%o A228650         b3 := k - (b1 + b2);
%o A228650         repeat
%o A228650         inc(b2); dec(b3);
%o A228650         if (b3 >= b2) and (b1 * b2 * b3 = p[b1] + p[b2] + p[b3]) then
%o A228650         begin
%o A228650           inc(n); inc(s);
%o A228650           writeln(n,' ',k);
%o A228650         end;
%o A228650       until (b3 <= b2) or (s > 0);
%o A228650     until (3 * b1 >= k) or (s > 0);
%o A228650   until k = 6000;
%o A228650 end.
%Y A228650 Cf. A228651, A228652, A228653.
%K A228650 nonn
%O A228650 1,1
%A A228650 _William Rex Marshall_, Aug 29 2013