A057655 -id:A057655 - cp's OEIS frontend

A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

1, 17, 129, 577, 1713, 3729, 6865, 12369, 21697, 33809, 47921, 69233, 101041, 136209, 174737, 231185, 306049, 384673, 469457, 579217, 722353, 876465, 1025649, 1220337, 1481521, 1733537, 1979713, 2306753, 2697537, 3087777, 3482913, 3959585, 4558737, 5155473

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A000143.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A000122, A000143, A001650, A046895, A055407, A055414, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341398, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 8)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..33);  # Alois P. Heinz, Feb 10 2021

nmax = 33; CoefficientList[Series[EllipticTheta[3, 0, x]^8/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[8, n], {n, 0, 33}] // Accumulate

from math import prod
from sympy import factorint
def A341397(n): return (sum((prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(m).items()) for m in range(1,n+1)))<<4)+1 # Chai Wah Wu, Jun 21 2024

G.f.: theta_3(x)^8 / (1 - x).

a(n^2) = A055414(n).

A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 3, 9, 27, 89, 333, 1341, 5449, 21697, 84663, 327829, 1275739, 5020457, 19964623, 79883141, 320317827, 1284656385, 5152761033, 20686311261, 83182322509, 335110196569, 1352277390001, 5463873556381, 22097867887045, 89441286136465, 362277846495883, 1468465431530457

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(n).

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Main diagonal of A122510.

Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]

a(n) = A122510(n,n).

a(n) ~ c / (sqrt(n) * r^n), where r = 0.241970723224463308846762732757915397312... (= radius of convergence A166952) and c = 0.716940866073606328... - Vaclav Kotesovec, Apr 14 2018

A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Original entry on oeis.org

1, 15, 99, 379, 953, 1793, 3081, 5449, 8893, 12435, 16859, 24419, 33659, 42115, 53203, 69779, 88273, 106081, 125821, 153541, 187981, 217437, 248741, 298469, 351277, 394691, 446939, 515259, 589307, 657683, 728803, 828259, 939223, 1029159, 1124023, 1260103

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A008451.

Index entries for sequences related to sums of squares

Cf. A000122, A001650, A008451, A046895, A055406, A055413, A057655, A117609, A122510, A175360, A175361, A302860, A341397, A341398, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 7)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 10 2021

nmax = 35; CoefficientList[Series[EllipticTheta[3, 0, x]^7/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[7, n], {n, 0, 35}] // Accumulate

my(q='q+O('q^(55))); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^7/(1-q)) \\ Joerg Arndt, Jun 21 2024

G.f.: theta_3(x)^7 / (1 - x).

a(n^2) = A055413(n).

A014198 Number of integer solutions to x^2 + y^2 <= n excluding (0,0).

Original entry on oeis.org

0, 4, 8, 8, 12, 20, 20, 20, 24, 28, 36, 36, 36, 44, 44, 44, 48, 56, 60, 60, 68, 68, 68, 68, 68, 80, 88, 88, 88, 96, 96, 96, 100, 100, 108, 108, 112, 120, 120, 120, 128, 136, 136, 136, 136, 144, 144, 144, 144, 148, 160, 160, 168, 176, 176, 176, 176, 176, 184, 184

Offset: 0

N. J. A. Sloane

a(32)/32 = 100/32 = 3.125; lim_{n->infinity} a(n)/n = Pi.

The terms of this sequence are four times the running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors. - Ant King, Mar 12 2013

			For n=2 the 8 solutions are (x,y) = (+-1,0), (0,+-1), (+-1,+-1).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 339

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Sum of Squares Function

Cf. A014200, A057655.

A014198 := proc(n)
    nops([ numtheory[thue]( abs( x^2+y^2) <= n, [ x, y ] ) ]);
end proc:
seq(A014198(n),n=0..60) ;

Prepend[SquaresR[2,#] &/@Range[59],0]//Accumulate (* Ant King, Mar 12 2013 *)

a(n)=local(j); j=sqrtint(n); sum(x=-j,j,sum(y=-j,j,x^2+y^2<=n))-1

from math import prod
from itertools import count, accumulate, islice
from sympy import factorint
def A014198_gen(): # generator of terms
    return accumulate(map(lambda n:prod(e+1 if p & 3 == 1 else (e+1) & 1 for p, e in factorint(n).items() if p > 2) << 2, count(1)),initial=0)
A014198_list = list(islice(A014198_gen(),30)) # Chai Wah Wu, Jun 28 2022

a(n) = 4*A014200(n).

a(n) = A057655(n)-1.

A036704 a(n)=number of Gaussian integers z=a+bi satisfying |z|<=n+1/2.

Original entry on oeis.org

1, 9, 21, 37, 69, 97, 137, 177, 225, 293, 349, 421, 489, 577, 665, 749, 861, 973, 1085, 1201, 1313, 1457, 1597, 1741, 1885, 2053, 2217, 2377, 2561, 2733, 2933, 3125, 3313, 3521, 3745, 3969, 4197, 4421, 4669, 4905, 5169, 5417

Offset: 0

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
Clive Tooth, A selection of a(n) values for n up to 10^9
Index entries for Gaussian integers and primes

Cf. A057655.

a(n) = A057655(n^2+n). - Robert Israel, Sep 29 2014

A038589 Sizes of successive clusters in hexagonal lattice A_2 centered at lattice point.

Original entry on oeis.org

1, 7, 7, 13, 19, 19, 19, 31, 31, 37, 37, 37, 43, 55, 55, 55, 61, 61, 61, 73, 73, 85, 85, 85, 85, 91, 91, 97, 109, 109, 109, 121, 121, 121, 121, 121, 127, 139, 139, 151, 151, 151, 151, 163, 163, 163, 163, 163, 169, 187, 187, 187, 199, 199, 199

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

			1 + 7*x + 7*x^2 + 13*x^3 + 19*x^4 + 19*x^5 + 19*x^6 + 31*x^7 + 31*x^8 + 37*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Benoit Cloitre, On the circle and divisor problems.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A004016, A014201, A038589, A038590.
Cf. A035019.
Cf. A057655 (for square lattice).

a[n_] := 1 + Sum[ Length[ {ToRules[ Reduce[ x^2 + x*y + y^2 == k, {x, y}, Integers] ]}], {k, 1, n}]; Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Feb 23 2012, after Neven Juric *)

a(n)=1+6*sum(k=0,n\3,(n\(3*k+1))-(n\(3*k+2)))

Partial sums of A004016.
Expansion of a(x) / (1 - x) in powers of x where a() is a cubic AGM theta function (cf. A004016). - Michael Somos, Aug 21 2012
Equals 1 + A014201(n). - Neven Juric, May 10 2010
a(n) = 1 + 6*Sum_{k=1..n/3} floor(n/(3k+1)) - floor(n/(3k+2)). a(n) is asymptotic to 2*(Pi/sqrt(3))*n. Conjecture: a(n) = 2*(Pi/sqrt(3))*n + O(n^(1/4 + epsilon)) as for the Gauss circle or Dirichlet divisor problems. - Benoit Cloitre, Oct 27 2012
a(n) = A014201(n) + 1. - Hugo Pfoertner, Nov 09 2023

A341398 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

Original entry on oeis.org

1, 19, 163, 835, 2869, 7189, 14581, 27253, 49861, 84663, 129303, 190071, 284055, 409335, 550455, 732855, 995241, 1312617, 1656153, 2077497, 2634777, 3300057, 4003641, 4804281, 5872665, 7129227, 8363307, 9784491, 11635755, 13670475, 15727755, 18066315, 20950491

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A008452.

Index entries for sequences related to sums of squares

Cf. A000122, A001650, A008452, A046895, A055408, A055415, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, A341399.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 9)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 10 2021

nmax = 32; CoefficientList[Series[EllipticTheta[3, 0, x]^9/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[9, n], {n, 0, 32}] // Accumulate

G.f.: theta_3(x)^9 / (1 - x).

a(n^2) = A055415(n).

A341399 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

Original entry on oeis.org

1, 21, 201, 1161, 4541, 12965, 29285, 58085, 110105, 198765, 327829, 503509, 765589, 1152509, 1642109, 2243069, 3083569, 4221529, 5551949, 7115789, 9166133, 11777333, 14763893, 18121973, 22316213, 27634481, 33512921, 39812441, 47674841, 57294401, 67510721, 78592961

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A000144.

Index entries for sequences related to sums of squares

Cf. A000122, A000144, A001650, A046895, A055409, A055416, A057655, A117609, A122510, A175360, A175361, A302860, A341396, A341397, A341398.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+2*add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 10)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..31);  # Alois P. Heinz, Feb 10 2021

nmax = 31; CoefficientList[Series[EllipticTheta[3, 0, x]^10/(1 - x), {x, 0, nmax}], x]
Table[SquaresR[10, n], {n, 0, 31}] // Accumulate

G.f.: theta_3(x)^10 / (1 - x).

a(n^2) = A055416(n).

A341198 Number of points on or inside the circle of radius n, as rasterized by the midpoint circle algorithm.

Original entry on oeis.org

1, 5, 21, 37, 61, 97, 129, 177, 221, 277, 349, 413, 489, 569, 657, 749, 845, 957, 1073, 1193, 1313, 1441, 1581, 1733, 1877, 2025, 2209, 2369, 2553, 2725, 2909, 3117, 3305, 3513, 3721, 3941, 4181, 4405, 4645, 4889, 5145, 5401, 5653, 5941, 6213, 6493, 6769, 7065

Offset: 0

Pontus von Brömssen, Feb 06 2021

nonn

The number of points on the rasterized circle itself (of radius n) is given by 4*A022846(n) for n > 0.

			In the figure below, the points on the rasterized circle of radius n are labeled with the number n. (Points without a label do not lie on any such circle.)
                9 9 9 9 9
            9 9 8 8 8 8 8 9 9
        9 9 8 8 7 7 7 7 7 8 8 9 9
      9 . 8 7 7 6 6 6 6 6 7 7 8 . 9
      9 8 7 . 6 5 5 5 5 5 6 . 7 8 9
    9 8 7 . 6 5 . 4 4 4 . 5 6 . 7 8 9
    9 8 7 6 5 4 4 3 3 3 4 4 5 6 7 8 9
  9 8 7 6 5 . 4 3 2 2 2 3 4 . 5 6 7 8 9
  9 8 7 6 5 4 3 2 . 1 . 2 3 4 5 6 7 8 9
  9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9
  9 8 7 6 5 4 3 2 . 1 . 2 3 4 5 6 7 8 9
  9 8 7 6 5 . 4 3 2 2 2 3 4 . 5 6 7 8 9
    9 8 7 6 5 4 4 3 3 3 4 4 5 6 7 8 9
    9 8 7 . 6 5 . 4 4 4 . 5 6 . 7 8 9
      9 8 7 . 6 5 5 5 5 5 6 . 7 8 9
      9 . 8 7 7 6 6 6 6 6 7 7 8 . 9
        9 9 8 8 7 7 7 7 7 8 8 9 9
            9 9 8 8 8 8 8 9 9
                9 9 9 9 9
Counting the points on or inside a circle of given radius, one obtains a(0)=1, a(1)=5, a(2)=21, a(3)=37, a(4)=61, a(5)=97, ...

Eric Weisstein's World of Mathematics, Gauss's Circle Problem
Wikipedia, Midpoint circle algorithm

First differences: A341199.

Cf. A022846, A055979, A057655, A373193.

def A341198(n):
  n2=n**2
  x=n
  y=A=0
  while y<=x:
    dx=x**2+(y+1)**2-n2-x>=0
    A+=x+(y!=0 and y!=x)*(x-2*y)+(dx and y==x-1)*(x-1)
    x-=dx
    y+=1
  return 4*A+1

a(n) == 1 (mod 4).

a(n) ~ Pi*n^2. More precisely, it is reasonable to expect that a(n) = Pi*n^2 + sqrt(8)*n + o(n), because there are Pi*n^2 + o(n) points in the disk x^2 + y^2 <= n^2 (Gauss's circle problem), all of which are inside the rasterized circle, and we can expect about half of the 4*sqrt(2)*n + O(1) points on the rasterized circle itself to be outside this disk (and there are no points between the disk and the rasterized circle).

A175363 Partial sums of A175362.

Original entry on oeis.org

1, 5, 9, 9, 9, 9, 9, 9, 13, 21, 21, 21, 21, 21, 21, 21, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 29, 37, 37, 37, 37, 37, 37, 37, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 53, 61, 61, 61, 61, 61, 61, 61

Offset: 0

R. J. Mathar, Apr 24 2010

nonn

Number of integer pairs (x,y) satisfying |x|^3+|y|^3 <= n, any -n <= x,y <=n. Cubic variant of A057655.

			a(1) = 5 counts (x,y) = (-1,0), (0,-1), (0,0), (0,1) and (1,0).

cp's OEIS Frontend

A341397 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

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A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

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A341396 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

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A014198 Number of integer solutions to x^2 + y^2 <= n excluding (0,0).

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A036704 a(n)=number of Gaussian integers z=a+bi satisfying |z|<=n+1/2.

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A038589 Sizes of successive clusters in hexagonal lattice A_2 centered at lattice point.

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A341398 Number of integer solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

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