A228650 - OEIS

%I #17 Jul 30 2022 08:19:14

%S 6,8,11,12,46,57,66,120,121,145,156,162,166,217,372,386,557,596,638,

%T 750,866,1025,1038,1201,1396,1857,2042,2081,2146,2263,2301,2452,2836,

%U 2900,2926,2991,3026,3053,3288,3368,3963,3970,4511,4656,5006,5492,5890,5952

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

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

%e 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 (Pascal) program a228650;

%o var

%o   p: array[1..6000] of int64;

%o   b1, b2, b3, k: int64;

%o   n, s: integer;

%o begin

%o   k:=0;

%o   repeat

%o     inc(k);

%o     p[k] := (k * (k - 1) * (k - 2)) div 6;

%o   until k = 6000;

%o   n := 0; k := 2;

%o   repeat

%o     inc(k); s := 0;

%o     b1 := 0;

%o     repeat

%o       inc(b1);

%o       b2 := b1 - 1;

%o         b3 := k - (b1 + b2);

%o         repeat

%o         inc(b2); dec(b3);

%o         if (b3 >= b2) and (b1 * b2 * b3 = p[b1] + p[b2] + p[b3]) then

%o         begin

%o           inc(n); inc(s);

%o           writeln(n,' ',k);

%o         end;

%o       until (b3 <= b2) or (s > 0);

%o     until (3 * b1 >= k) or (s > 0);

%o   until k = 6000;

%o end.

%Y Cf. A228651, A228652, A228653.

%K nonn

%O 1,1

%A _William Rex Marshall_, Aug 29 2013