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A309573 a(n) is the sum of lattice points enumerated by the square number spiral falling on the circumference of circles centered at the origin of radii n.

%I A309573 #34 Feb 16 2025 08:33:55
%S A309573 0,16,64,144,256,912,576,784,1024,1296,3648,1936,2304,7312,3136,8208,
%T A309573 4096,11824,5184,5776,14592,7056,7744,8464,9216,41232,29248,11664,
%U A309573 12544,27568,32832,15376,16384,17424,47296,44688,20736,61104,23104,65808,58368,78096,28224,29584,30976,73872
%N A309573 a(n) is the sum of lattice points enumerated by the square number spiral falling on the circumference of circles centered at the origin of radii n.
%C A309573 For this sequence the square spiral begins with 0 and is the second illustration in the comments of A317186, where 0 is the origin of our circles.
%C A309573 a(n) >= A001107(n) + A033991(n) + A007742(n) + A033954(n).
%C A309573 a(n) = A016802(n) iff A046109(n) = 4.
%C A309573 a(n) = A016802(n) iff n <> k * A002144(m), k,m >= 1.
%C A309573 a(n) is congruent to 0 mod 16 and is the sum of one or more terms of A016802.
%C A309573 Conjecture: a(n) is a term of A277699 iff a(n)/16 = A277699(n).
%H A309573 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CircleLatticePoints.html">Circle Lattice Points</a>
%e A309573 16 is a term because 16 = 16*(1)^2.
%e A309573 912 is a term because 912 = 16*(5)^2 + (2*(16*(4)^2)).
%e A309573 41232 is a term because 41232 = 16*(25)^2 + (2*((16*(24)^2) + (16*(20)^2))).
%o A309573 (PARI) Tb(n) = {return(16 * n * n)}
%o A309573 llsum(n) = {my(x=0); for (i = 1, n - 2, for (ii = i+1, n - 1, if(n*n == (ii*ii) + (i*i), x+=(2 * Tb(ii))))); return(x)}
%o A309573 Tx(n) = {my(x=0); forprimestep(x = 5, n, 4, if(n%x==0, return(llsum(n))))}
%o A309573 Tn(n) = {for (i = 0, n, print1(Tb(i) + Tx(i), ", "))}
%o A309573 Tn(45)
%Y A309573 Cf. A001107, A002144, A007742, A016802, A033954, A033991, A046109, A277699, A317186.
%K A309573 nonn
%O A309573 0,2
%A A309573 _Torlach Rush_, Aug 08 2019