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

0,2

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).

Links

Table of n, a(n) for n=0..45.

Eric Weisstein's World of Mathematics, Circle Lattice Points

Example

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))).

Prog

(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)

Crossrefs

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

Sequence in context: A072128 A277016 A016802 * A205064 A102860 A136264

Adjacent sequences: A309570 A309571 A309572 * A309574 A309575 A309576

Keyword

nonn

Author

Torlach Rush, Aug 08 2019

Status

approved