OFFSET
1,1
COMMENTS
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).
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..1000
Byeong-Kweon Oh and Zhi-Wei Sun, Mixed sums of square and triangular numbers (III), Journal of Number Theory 129:4, (2009), pp. 964-969.
EXAMPLE
MATHEMATICA
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
PROG
(PARI)
N=10^5; /* upper bound */
x='x+O('x^N);
S=2*ceil(sqrt(N));
tr=sum(n=0, S, x^(n*(n+1)/2)); /* triangular incl. zero */
sq=sum(n=1, S, x^((2*n-1)^2)); /* odd squares */
f=tr^2*sq + 't /* symbol t to have vector aligned */
v=Vec(f);
for(n=1, #v, if(v[n]==0, print1(n-1, ", ")));
/* Joerg Arndt, Jul 06 2011 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ant King, Jun 22 2011
STATUS
approved