OFFSET
1,2
COMMENTS
We have A000330(n) = 1 + 2^2 + ... + n^2 and A000384(m) = 1 - 2^2 + 3^2 -+ ... + (2m-1)^2, so the terms of this sequence are numbers that are a product, in at least two ways, of a partial sum of squares times a (positive) partial sum of squares with alternating signs (with + for odd terms; cf. A306122 for the opposite convention).
The initial a(1) = 0 is added for completeness.
Below 10^8, only the two numbers 2748900 and 5705700 have three representations of the given form, and none has four.
LINKS
Geoffrey Campbell, Integer solutions of (1²-2²+3²-...+(2a-1)²) × (1²+2²+3²+...+b²) = (1²-2²+3²-...+(2c-1)²) × (1²+2²+3²+...+d²) where a ≠ c and b ≠ d, Number Theory group on LinkedIn, June 2018.
PROG
(PARI) {my(L=10^6, A384(a)=a*(2*a-1), A330(b)=(b+1)*b*(2*b+1)/6, A=S=[]); for(b=1, sqrtnint(L\A384(1)\3, 3), for(a=1, oo, if( setsearch(S, t=A384(a)*A330(b)), A=setunion(A, [t]), t>L&&next(2); S=setunion(S, [t])))); A}
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey B. Campbell and M. F. Hasler, Jul 03 2018
STATUS
approved