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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.
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%I #33 Oct 08 2019 22:19:29

%S 0,16,64,144,256,912,576,784,1024,1296,3648,1936,2304,7312,3136,8208,

%T 4096,11824,5184,5776,14592,7056,7744,8464,9216,41232,29248,11664,

%U 12544,27568,32832,15376,16384,17424,47296,44688,20736,61104,23104,65808,58368,78096,28224,29584,30976,73872

%N 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 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 a(n) >= A001107(n) + A033991(n) + A007742(n) + A033954(n).

%C a(n) = A016802(n) iff A046109(n) = 4.

%C a(n) = A016802(n) iff n <> k * A002144(m), k,m >= 1.

%C a(n) is congruent to 0 mod 16 and is the sum of one or more terms of A016802.

%C Conjecture: a(n) is a term of A277699 iff a(n)/16 = A277699(n).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircleLatticePoints.html">Circle Lattice Points</a>

%e 16 is a term because 16 = 16*(1)^2.

%e 912 is a term because 912 = 16*(5)^2 + (2*(16*(4)^2)).

%e 41232 is a term because 41232 = 16*(25)^2 + (2*((16*(24)^2) + (16*(20)^2))).

%o (PARI) Tb(n) = {return(16 * n * n)}

%o 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 Tx(n) = {my(x=0); forprimestep(x = 5, n, 4, if(n%x==0, return(llsum(n))))}

%o Tn(n) = {for (i = 0, n, print1(Tb(i) + Tx(i), ", "))}

%o Tn(45)

%Y Cf. A001107, A002144, A007742, A016802, A033954, A033991, A046109, A277699, A317186.

%K nonn

%O 0,2

%A _Torlach Rush_, Aug 08 2019