%I A127690
%S A127690 3,4,12,84,3612,6526884,21300113901612,226847426110843688722000884,
%T A127690 25729877366557343481074291996721923093306518970391612
%N A127690 a(1)=3; for n>1, a(n) is least number such a(1)^2+...+a(n)^2 is a square 
               and is equal to (1+a(n))^2.
%D A127690 Sierpinski W., 1959 Toria Liczb (=Theory of Numbers) Part II. Monografie 
               Matematyczne Tom 38. PWN Warszawa, pp. 1-487 (p.76).
%F A127690 Conjecture: a(n)=A053630(n-1)-1 for n>=2. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Apr 23 2007
%e A127690 a(2)=4 because (3^2+4^2=5^2) and (4+1=5), a(3)=12 because (3^2+4^2+12^2=13^2) 
               and (12+1=13) a(5)= 3612 because (3^2+4^2+12^2+84^2+3612^2=3613^2) 
               and (3612+1=3613) etc.
%t A127690 a = {3}; For[k = 1 + a[[Length[a]]], Length[a] < 5, While[ ! ((IntegerQ[Sqrt[(k)^2 
               + Sum[(a[[t]])^2, {t, 1, Length[a]}]]]) && (Sqrt[(k)^2 + Sum[(a[[t]])^2, 
               {t, 1, Length[a]}]] == k + 1)), k++ ]; AppendTo[a, k]]; a
%t A127690 a = {3}; For[k = 1 + a[[Length[a]]], Length[a] < 12, s2 = Plus @@ (a^2); 
               t = Reduce[{y^2 + s2 == (y + 1)^2}, y, Integers]; t = t /. {Equal 
               -> Rule}; k = y /. t; AppendTo[a, k]]; a (* Daniel Huber *)
%Y A127690 Cf. A018930, A127689, A127691.
%Y A127690 Sequence in context: A059792 A018930 A127689 this_sequence A092417 A071543 
               A127611
%Y A127690 Adjacent sequences: A127687 A127688 A127689 this_sequence A127691 A127692 
               A127693
%K A127690 nonn
%O A127690 1,1
%A A127690 Artur Jasinski (grafix(AT)csl.pl), Jan 23 2007, Jan 29 2007