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.

0, 16, 64, 144, 256, 912, 576, 784, 1024, 1296, 3648, 1936, 2304, 7312, 3136, 8208, 4096, 11824, 5184, 5776, 14592, 7056, 7744, 8464, 9216, 41232, 29248, 11664, 12544, 27568, 32832, 15376, 16384, 17424, 47296, 44688, 20736, 61104, 23104, 65808, 58368, 78096, 28224, 29584, 30976, 73872

Offset: 0

Author

Torlach Rush, Aug 08 2019

Keywords

• nonn

Comments

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.

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

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

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

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

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

Examples

			16 is a term because 16 = 16*(1)^2.
912 is a term because 912 = 16*(5)^2 + (2*(16*(4)^2)).
41232 is a term because 41232 = 16*(25)^2 + (2*((16*(24)^2) + (16*(20)^2))).
		

Links

• Eric Weisstein's World of Mathematics, Circle Lattice Points

Crossrefs

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

Programs

PARI

Tb(n) = {return(16 * n * n)}
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)}
Tx(n) = {my(x=0); forprimestep(x = 5, n, 4, if(n%x==0, return(llsum(n))))}
Tn(n) = {for (i = 0, n, print1(Tb(i) + Tx(i), ", "))}
Tn(45)