%I #12 Jul 28 2016 21:15:51
%S 15,21,28,51,55,64,70,75,78,91,96,100,111,112,117,126,135,136,141,144,
%T 145,148,154,156,165,175,176,186,189,195,201,204,216,232,235,238,246,
%U 255,256,285,286,288,291,297,300,306,315,316,321,322,324,330,333,336
%N Integers n that are k-gonal for precisely 3 distinct values of k, where k >= 3.
%C See A177025 for number of ways a number can be represented as a polygonal number.
%H Chai Wah Wu, <a href="/A195527/b195527.txt">Table of n, a(n) for n = 1..10000</a>
%e 21 is in the sequence because it is a triangular number (A000217), an octagonal number (A000567) and an icosihenagonal number (A051873).
%t data1=Reduce[1/2 n (n(k-2)+4-k)== # && k>=3 && n>0, {k,n}, Integers]&/@Range[336]; data2=If[Head[#]===And, 1, Length[#]] &/@data1; data3=DeleteCases[Table[If[data2[[k]]==3, k], {k, 1, Length[data2]}], Null]
%o (Python)
%o A195527_list = []
%o for m in range(1,10**4):
%o n, c = 3, 0
%o while n*(n+1) <= 2*m:
%o if not 2*(n*(n-2) + m) % (n*(n - 1)):
%o c += 1
%o if c > 2:
%o break
%o n += 1
%o if c == 2:
%o A195527_list.append(m) # _Chai Wah Wu_, Jul 28 2016
%Y Cf. A000217, A000567, A051873, A177025, A177029.
%K nonn
%O 1,1
%A _Ant King_, Sep 21 2011