A081018: Pythagorean triple interpretation

[Nick Hobson]

Hi Seqfans,

Another interpretation of this sequence is: non-negative integers k such  
that (k + 1)^2 + (2k)^2 is a perfect square.  So, apart from a(0) = 0,  
a(n) + 1 and 2a(n) form the legs of a pythagorean triple.

Nick



Here's a sequence that I'm interested in computing, assuming that it exists.

Consider all of the positive, constant-spaced integers:

1 2 3 4 5
1 3 5 7 9
2 4 6 8 10
1 4 7 10 13
2 5 8 11 14
3 6 9 12 15
etc.

The figurate numbers are the sums of rows starting with 1. The other rows
have various names, but are constructed similarly thus I'm including them
here. The question is are there any composite numbers greater than 6 which
cannot be expressed as the sum of one of these rows minus the initial value?
Some examples:

2+3+4+5 = 14 (or 1+2+3+4+5 - 1)
3+5+7 = 15 (or 1+3+5+7 - 1)
5+8+11+14+17 = 55 (or 2+5+8+11+14+17 - 2)

Thus 14, 15, and 55 are expressible in this way. My initial thought is that
all composites have at least one such representation, thus my sequence
doesn't exist. Any ideas on how I would go about constructing this sequence?
Thanks in advance for your help!