%I #12 Mar 30 2012 18:41:05

%S 6,42,72,156,210,342,420,702,930,1056,1332,1806,1980,2352,2550,2970,

%T 3192,3906,4692,5256,5550,6162,7140,7482,8190,8556,9312,9702,10506,

%U 12210,13110,13572,14520,16512,17556,18090,19182,19740,20880,21462,23256,24492,25122,26406,28392,30450,31152,33306,34782,35532,37830

%N Integers that do not have a partition into a sum of an odd square and two (not necessarily distinct) triangular numbers.

%C Oh & Sun have proved that a natural number cannot be partitioned into a sum of an odd square and two triangular numbers if and only if it is a pronic number A002378 (m) such that 2m+1 does not have any prime divisors that are congruent to 3 (mod 4).

%H Donovan Johnson, <a href="/A191764/b191764.txt">Table of n, a(n) for n = 1..1000</a>

%H Byeong-Kweon Oh and Zhi-Wei Sun, <a href="http://arxiv.org/abs/0804.3750">Mixed sums of square and triangular numbers (III)</a>, Journal of Number Theory 129:4, (2009), <a href="http://dx.doi.org/10.1016/j.jnt.2008.10.002">pp. 964-969</a>.

%e The fifth integer that does not have a partition into a sum of an odd square and two triangular numbers is 210. Hence a(5)=210. Similarly, 21 is not in the sequence as it has a unique representation as A000290(3)+A000217(3)+A000217(3)

%t data=Length[FindInstance[(2x+1)^2+1/2 y (y+1)+1/2 z (z+1)==# && 0<=x<=# && 0<=y<=# && 0<=z<=#,{x,y,z},Integers]]&/@Range[10000];Position[data,0]//Flatten

%o (PARI)

%o N=10^5; /* upper bound */

%o x='x+O('x^N);

%o S=2*ceil(sqrt(N));

%o tr=sum(n=0,S,x^(n*(n+1)/2)); /* triangular incl. zero */

%o sq=sum(n=1,S,x^((2*n-1)^2)); /* odd squares */

%o f=tr^2*sq + 't /* symbol t to have vector aligned */

%o v=Vec(f);

%o for(n=1,#v,if(v[n]==0,print1(n-1,", ")));

%o /* Joerg Arndt, Jul 06 2011 */

%Y Cf. A002378, A016754, A000217.

%K nonn,easy

%O 1,1

%A _Ant King_, Jun 22 2011