%I
%S 0,1,4,8,9,16,17,25,36,40,49,52,64,72,73,80,81,89,97,100,116,121,136,
%T 144,145,148,169,180,193,196,225,232,233,241,244,256,260,288,289,292,
%U 305,313,324,337,360,361,369,388,400,404,409,424,441,449,457
%N Integers k such that both k and k+1 are the sum of 2 squares.
%C Equivalently, nonnegative k such that k*(k+1) is the sum of two squares.
%C Also, nonnegative k such that k*(k+1)/2 is the sum of two squares. This follows easily from the "sum of two squares theorem": x is the sum of two (nonnegative) squares iff its prime factorization does not contain p^e where p == 3 (mod 4) and e is odd.  _Robert Israel_, Mar 26 2018
%C Trivially, sequence includes all positive squares.
%H David A. Corneth, <a href="/A140612/b140612.txt">Table of n, a(n) for n = 1..10000</a>
%e 40 = 6^2 + 2^2, 41 = 5^2 + 4^2, so 40 is in the sequence.
%t (*M6*) A1 = {}; Do[If[SquaresR[2, n (n + 1)/2] > 0, AppendTo[A1, n]], {n, 0, 1500}]; A1
%t Join[{0}, Flatten[Position[Accumulate[Range[500]], _?(SquaresR[2, #]> 0&)]]] (* _Harvey P. Dale_, Jun 07 2015 *)
%o (MAGMA) [k:k in [0..460] forall{k+a: a in [0,1]NormEquation(1, k+a) eq true}]; // _Marius A. Burtea_, Oct 08 2019
%Y Cf. A000217, A000290, A000404, A001481, A002378, A050795, A073613.
%K nonn,easy
%O 1,3
%A _Franklin T. AdamsWatters_, May 19 2008
%E a(1)=0 prepended and edited by _Max Alekseyev_, Oct 08 2019
