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A177029 Numbers that have exactly two different representations as polygonal numbers.

%I A177029 #15 Jul 28 2016 21:15:40
%S A177029 6,9,10,12,16,18,22,24,25,27,30,33,34,35,39,40,42,46,48,49,52,54,57,
%T A177029 58,60,63,65,69,72,76,82,84,85,87,88,90,92,93,94,95,99,102,106,108,
%U A177029 114,115,118,121,123,124,125,129,130,132,133,138,142,147,150,155,159,160,162,166,168
%N A177029 Numbers that have exactly two different representations as polygonal numbers.
%C A177029 Numbers that have only A177025(.)=1 representation are listed by A090467.
%H A177029 Chai Wah Wu, <a href="/A177029/b177029.txt">Table of n, a(n) for n = 1..10000</a>
%F A177029 {m: A177025(m)=2}.
%e A177029 6 is a triangular and a hexagonal number, but is not any other k-gonal number.
%o A177029 (PARI) lista(nn) = {for (n=1, nn, if (sum(k=3, n, ispolygonal(n, k)) == 2, print1(n, ", ")););} \\ _Michel Marcus_, Mar 25 2015
%o A177029 (Python)
%o A177029 A177029_list = []
%o A177029 for m in range(1,10**4):
%o A177029     n, c = 3, 0
%o A177029     while n*(n+1) <= 2*m:
%o A177029         if not 2*(n*(n-2) + m) % (n*(n - 1)):
%o A177029             c += 1
%o A177029             if c > 1:
%o A177029                 break
%o A177029         n += 1
%o A177029     if c == 1:
%o A177029         A177029_list.append(m) # _Chai Wah Wu_, Jul 28 2016
%Y A177029 Cf. A177025, A090467, A177028, A000217, A000326, A000384, A000566, A000567, A176949, A176948, A176774, A176744, A176747, A176775, A175873, A176874.
%K A177029 nonn
%O A177029 1,1
%A A177029 _Vladimir Shevelev_, May 01 2010
%E A177029 Extended by _R. J. Mathar_, Aug 15 2010