[quote="Annunaki"]hmm.. i will look at it a little bit later.. but for now it seems that it may be something good.. Hovewer we need to find a way to include 0th element..
and in general case i need only to find soonest square not all of them.. but of course this mehod still looks good for now..[/quote]

I don't see a way to include an arbitrary zeroth element into this approach - is there perhaps some relationship between your zeroth element and your initial element we could exploit?

I haven't thought too much about the "soonest" issue, but because f = 2a-1 and a = d*(p^2+q^2), it should be easy to select the values of d, p, and q that give the smallest a and therefor the smallest f.

i fear there are no more relationship between oth elment and first element.. the only thing I can say about 0th element that it will be allways at least by 3 lesser than fisrt element.. thus the 0th<1st-2
I think the only way to include 0th element is to find a squence that has exactly the same number of elements, but the sum is allways largen than that of the orginal sequenece by 0th element
for example if:
sequence is
8, 21, 23, 25 ,27
we can shiwtch it to 23,25,27,29..
but that is only in casw with 4 elements but i still think that it can be done..
only i still cant figure how..

may be a proof to hte statement that every Pythagoreon triple can be formed from natural p and q if sides are - 2pq; p^2+q^2; p^2-q^2; could help us..
thus may be if we know on what is based this statement we can step a little bit higher..

hmm i think i noticed something..
and i can include 0th element..
can you check easely when some triple will be Pythagoreon triple..
for example
if i know two sides of that triple.. and they grow in progression..
Can i know when natural Pythagoreon triple will be posible for example i know.. that
two sides of triple are
26; 25
28; 25
30; 25
etc.. can i tell when triple of Pythagorian will be possible.. According to what i can read it is posible even very posible.. but that is too easy.. that is why I can`t believe in that.. I will think about it again..

hmh .. i think i found a bug in my previous post..
that is not so easy..
i dont need to find only when it will be Pythagoreon triple..
i Actually need to know if pythagoreon triple would be posible if I add 0th element:(
in that case it would be.. 0th=14
26^2-25^2 +14
28^2-25^2 +14
30^2-25^2 +14
etc. notice that 30^2-25^2+14 gives us answer as it is 289=17^2..
but can we tell it somehow more efectively..