A210639 -id:A210639 - cp's OEIS frontend

A210641 A117609(n)-A210639(n): Difference between number of lattice points in the ball x^2+y^2+z^2 <= n and the volume of this ball rounded to the nearest integer.

1, 3, 7, 5, -1, 10, 19, 3, -2, 10, 15, 18, 5, 7, 32, 8, -11, 11, 21, 18, 14, 34, 29, -1, -7, -9, 32, 31, -2, 37, 51, 16, -7, 5, 17, 28, 20, 6, 40, 1, -15, 41, 49, 32, 14, 45, 50, 7, -28, -18, 22, 25, 4, 31, 81, 34, 36, 36, 13, 37, -12, 11, 58, 8, -36, 10, 55

Offset: 0

M. F. Hasler, Mar 26 2012

sign

Record values are listed in A000223, and A000092 gives the corresponding indices. Strictly speaking, these are defined using the absolute values, but it appears they always occur at positive elements.

a[n_] := Sum[SquaresR[3, k], {k, 0, n}] - Round[(4/3)*Pi*n^(3/2)]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 04 2016 *)

PARI
```
A210641(n)=A117609(n)-A210639(n)
```

A117609 Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.

Original entry on oeis.org

1, 7, 19, 27, 33, 57, 81, 81, 93, 123, 147, 171, 179, 203, 251, 251, 257, 305, 341, 365, 389, 437, 461, 461, 485, 515, 587, 619, 619, 691, 739, 739, 751, 799, 847, 895, 925, 949, 1021, 1021, 1045, 1141, 1189, 1213, 1237, 1309, 1357, 1357, 1365, 1419, 1503

Offset: 0

John L. Drost, Apr 06 2006

nonn

			a(2) = 1 + 6 + 12 = 19, since (0,0,0) and (0, 0, +-1) and cyclic permutations (for a total of 6 points), and +-(0, 1, +-1) and cyclic permutations (for a total 12 points) are inside or on x^2 + y^2 + z^2 = 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
S. K. K. Choi, A. V. Kumchev and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.

Partial sums of A005875.
Cf. A000605 (number of points of norm <= n in cubic lattice).
Cf. A210639, A000092 and references therein.
Cf. A057655.

Table[Sum[SquaresR[3,k], {k,0,n}], {n,0,50}] (* T. D. Noe, Apr 08 2006, revised Sep 27 2011 *)

A117609(n)=sum(x=0,sqrtint(n),(sum(y=1,sqrtint(t=n-x^2),1+2*sqrtint(t-y^2))*2+sqrtint(t)*2+1)*2^(x>0)) \\ M. F. Hasler, Mar 26 2012

q='q+O('q^66); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^3/(1-q)) /* Joerg Arndt, Apr 08 2013 */

# uses Python code for A057655
from math import isqrt
def A117609(n): return A057655(n)+(sum(A057655(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 23 2024

a(n) ~ (4/3)*Pi*n^(3/2) ~ A210639(n).
a(n) = A122510(3,n). - R. J. Mathar, Apr 21 2010
G.f.: T3(q)^3/(1-q) where T3(q) = 1 + 2*Sum_{k>=1} q^(k^2). - Joerg Arndt, Apr 08 2013
a(n^2) = A000605(n). - R. J. Mathar, Aug 03 2025

A000092 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record.

Original entry on oeis.org

1, 2, 5, 6, 14, 21, 29, 30, 54, 90, 134, 155, 174, 230, 234, 251, 270, 342, 374, 461, 494, 550, 666, 750, 810, 990, 1890, 2070, 2486, 2757, 2966, 3150, 3566, 3630, 4554, 4829, 5670, 5750, 8154, 8382, 8774, 8910, 10350, 10710, 15734, 15750, 16302, 17550

Offset: 1

N. J. A. Sloane

nonn

Indices n for which A210641(n) = A117609(n) - A210639(n) yields record values (in absolute value). - M. F. Hasler, Mar 26 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seth A. Troisi, Table of n, a(n) for n = 1..131
W. C. Mitchell, The number of lattice points in a k-dimensional hypersphere, Math. Comp., 20 (1966), 300-310.
Seth A. Troisi, Python program

Cf. A000323, A000036, A000099, A000413, A000223.

P[n_] := Sum[SquaresR[3, k], {k, 0, n}] - Round[(4/3)*Pi*n^(3/2)]; record = 0; A000092 = Reap[For[n=1, n <= 2*10^4, n++, If[(p = Abs[P[n]]) > record, record = p; Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 04 2016, after M. F. Hasler *)

m=0; for(n=1,1e4, if(m+0A210641(n)),m),print1(n",")))  /* Start with n=0 to print the initial 0. */ \\ M. F. Hasler, Mar 26 2012

Revised Jun 28 2005

A000223 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives (nearest integer to, I believe) P(A000092(n)).

Original entry on oeis.org

3, 7, 10, 19, 32, 34, 37, 51, 81, 119, 122, 134, 157, 160, 161, 174, 221, 252, 254, 294, 305, 309, 364, 371, 405, 580, 682, 734, 756, 763, 776, 959, 1028, 1105, 1120, 1170, 1205, 1550, 1570, 1576, 1851, 1930, 2028, 2404, 2411, 2565, 2675, 2895, 2905, 2940, 3133, 3211, 3240, 3428

Offset: 1

N. J. A. Sloane

nonn

Record values of (absolute values of) A210641 = A117609-A210639. It appears that the records occur always at positive elements of that sequence. (One could add an initial a(0)=1.) - M. F. Hasler, Mar 26 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seth A. Troisi, Table of n, a(n) for n = 1..131
W. C. Mitchell, The number of lattice points in a k-dimensional hypersphere, Math. Comp., 20 (1966), 300-310.

Cf. A000323, A000036, A000092, A000413, A000099.

nmax = 3*10^4; P[n_] := Sum[SquaresR[3, k], {k, 0, n}] - Round[(4/3)*Pi* n^(3/2)]; record = 0; A000223 = Reap[For[n = 1, n <= nmax, n++, If[(p = Abs[pn = P[n]]) > record, record = p; Print[pn]; Sow[pn]]]][[2, 1]] (* Jean-François Alcover, Feb 05 2016 *)

m=0;for(n=0,1e4, mA210641(n)) & print1(m=A210641(n)",")) /* This would print a negative value in case the record in absolute value occured for A117609(n)<A210639(n), which does not happen for n<10^4. */ \\ M. F. Hasler, Mar 26 2012

a(n) = |A210641(A000092(n))|. - M. F. Hasler, Mar 26 2012

Revised Jun 28 2005

cp's OEIS Frontend

A210641 A117609(n)-A210639(n): Difference between number of lattice points in the ball x^2+y^2+z^2 <= n and the volume of this ball rounded to the nearest integer.

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A117609 Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.

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A000092 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record.

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A000223 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives (nearest integer to, I believe) P(A000092(n)).

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