A000606 -id:A000606 - cp's OEIS frontend

A224212 Number of nonnegative solutions to x^2 + y^2 <= n.

1, 3, 4, 4, 6, 8, 8, 8, 9, 11, 13, 13, 13, 15, 15, 15, 17, 19, 20, 20, 22, 22, 22, 22, 22, 26, 28, 28, 28, 30, 30, 30, 31, 31, 33, 33, 35, 37, 37, 37, 39, 41, 41, 41, 41, 43, 43, 43, 43, 45, 48, 48, 50, 52, 52, 52, 52, 52, 54, 54, 54, 56, 56, 56, 58, 62, 62, 62, 64

Offset: 0

Alex Ratushnyak, Apr 01 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000925 (first differences).
Cf. A000606 (number of nonnegative solutions to x^2 + y^2 + z^2 <= n).
Cf. A057655 (number of integer solutions to x^2 + y^2 <= n).

nn = 68; t = Table[0, {nn}]; Do[d = x^2 + y^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)

for n in range(99):
  k = 0
  for x in range(99):
    s = x*x
    if s>n: break
    for y in range(99):
        sy = s + y*y
        if sy>n: break
        k+=1
  print(str(k), end=', ')

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^2. - Ilya Gutkovskiy, Mar 14 2017

A002102 Number of nonnegative solutions to x^2 + y^2 + z^2 = n.

Original entry on oeis.org

1, 3, 3, 1, 3, 6, 3, 0, 3, 6, 6, 3, 1, 6, 6, 0, 3, 9, 6, 3, 6, 6, 3, 0, 3, 9, 12, 4, 0, 12, 6, 0, 3, 6, 9, 6, 6, 6, 9, 0, 6, 15, 6, 3, 3, 12, 6, 0, 1, 9, 15, 6, 6, 12, 12, 0, 6, 6, 6, 9, 0, 12, 12, 0, 3, 18, 12, 3, 9, 12, 6, 0, 6, 9, 18, 7, 3, 12, 6, 0, 6, 15, 9, 9, 6, 12, 15, 0, 3, 21, 18, 6, 0, 6

Offset: 0

N. J. A. Sloane

nonn

A. Das and A. C. Melissinos, Quantum Mechanics: A Modern Introduction, Gordon and Breach, 1986, p. 48.
H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
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).

T. D. Noe, Table of n, a(n) for n = 0..10000

First differences of A000606.

a[n_] := Module[{x, y, z, c}, For[x=c=0, x^2<=n, x++, For[y=0, x^2+y^2<=n, y++, If[IntegerQ[Sqrt[n-x^2-y^2]], c++ ]]]; c]
CoefficientList[Series[Sum[q^n^2, {n, 0, 12}], {q, 0, 150}]^3, q]

Vec(sum(k=0,9,x^(k^2),O(x^100))^3) \\ Charles R Greathouse IV, Jun 13 2012

Coefficient of q^k in (1/8)*(1 + theta_3(0, q))^3, or coefficient of q^n in (1 + q + q^4 + q^9 + q^16 + q^25 + q^36 + q^49 + q^64 + ...)^3.

More terms from Dean Hickerson, Oct 07 2001

A224213 Number of nonnegative solutions to x^2 + y^2 + z^2 + u^2 <= n.

Original entry on oeis.org

1, 5, 11, 15, 20, 32, 44, 48, 54, 70, 88, 100, 108, 124, 148, 160, 165, 189, 219, 235, 253, 281, 305, 317, 329, 357, 399, 427, 439, 475, 523, 539, 545, 581, 623, 659, 688, 716, 764, 792, 810, 858, 918, 946, 970, 1030, 1078, 1102, 1110, 1154, 1226, 1274, 1304, 1352

Offset: 0

Alex Ratushnyak, Apr 01 2013

nonn

Cf. A014110 (first differences).
Cf. A224212 (number of nonnegative solutions to x^2 + y^2 <= n).
Cf. A000606 (number of nonnegative solutions to x^2 + y^2 + z^2 <= n).
Cf. A046895 (number of integer solutions to x^2 + y^2 + z^2 + u^2 <= n).

nn = 50; t = Table[0, {nn}]; Do[d = x^2 + y^2 + z^2 + u^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}, {z, 0, nn}, {u, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)

for n in range(99):
  k = 0
  for x in range(99):
    s = x*x
    if s>n: break
    for y in range(99):
        sy = s + y*y
        if sy>n: break
        for z in range(99):
            sz = sy + z*z
            if sz>n: break
            for u in range(99):
              su = sz + u*u
              if su>n: break
              k+=1
  print(str(k), end=', ')

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^4. - Ilya Gutkovskiy, Mar 14 2017

A000604 Number of nonnegative solutions to x^2 + y^2 + z^2 <= n^2.

Original entry on oeis.org

1, 4, 11, 29, 54, 99, 163, 239, 344, 486, 648, 847, 1069, 1355, 1680, 2046, 2446, 2911, 3443, 4022, 4662, 5395, 6145, 6998, 7913, 8913, 10006, 11194, 12437, 13751, 15216, 16710, 18361, 20123, 21950, 23919, 25956, 28150, 30415, 32876, 35385, 38049, 40876

Offset: 0

N. J. A. Sloane

nonn

H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
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).

T. D. Noe, Table of n, a(n) for n=0..500

Column k=3 of A302998.

Cf. A000606.

a[n_] := Sum[Boole[x^2 + y^2 + z^2 <= n^2], {x, 0, n}, {y, 0, Sqrt[n^2 - x^2]}, {z, 0, Sqrt[n^2 - x^2 - y^2]}]; A000604 = Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 100}] (* Jean-François Alcover, Feb 10 2016 *)

a(n) = [x^(n^2)] (1 + theta_3(x))^3/(8*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

More terms from David W. Wilson, May 22 2000

A302862 a(n) = [x^n] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 8, 20, 57, 160, 422, 1076, 2780, 7449, 20462, 56348, 153909, 418268, 1139703, 3126068, 8618611, 23801146, 65708424, 181391905, 501296216, 1387834518, 3848187985, 10680579812, 29660831057, 82415406493, 229156296047, 637659848888, 1775648562970, 4947475298595

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_n)^2 <= n.

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

Cf. A000606, A003059, A010052, A224212, A224213, A287617, A302860, A302863.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n}], {n, 0, 30}]

A341400 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n.

Original entry on oeis.org

1, 6, 16, 26, 36, 57, 87, 107, 122, 157, 207, 247, 277, 322, 392, 452, 482, 537, 637, 717, 773, 863, 973, 1053, 1113, 1203, 1343, 1473, 1553, 1668, 1858, 1998, 2053, 2173, 2373, 2543, 2673, 2818, 3018, 3218, 3338, 3483, 3753, 3973, 4113, 4344, 4634, 4834, 4944, 5139, 5449

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A038671.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A038671, A055404, A055411, A175360, A224212, A224213, A302862, A341401, A341402.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 5)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 10 2021

nmax = 50; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^5/(32 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^5 / (32 * (1 - x)).

a(n^2) = A055404(n).

A341401 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n.

Original entry on oeis.org

1, 7, 22, 42, 63, 99, 160, 220, 265, 337, 457, 577, 672, 792, 978, 1178, 1319, 1469, 1739, 2039, 2255, 2507, 2882, 3242, 3513, 3819, 4269, 4769, 5159, 5555, 6181, 6841, 7246, 7666, 8401, 9181, 9763, 10363, 11188, 12108, 12828, 13434, 14394, 15534, 16359, 17211, 18477, 19677

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045848.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045848, A055405, A055412, A175361, A224212, A224213, A302862, A341400, A341402.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 6)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..47);  # Alois P. Heinz, Feb 10 2021

nmax = 47; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^6/(64 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^6 / (64 * (1 - x)).

a(n^2) = A055405(n).

A341402 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Original entry on oeis.org

1, 8, 29, 64, 106, 169, 281, 422, 548, 702, 961, 1276, 1556, 1864, 2326, 2893, 3390, 3852, 4545, 5455, 6253, 7002, 8080, 9361, 10453, 11496, 12903, 14618, 16194, 17643, 19589, 22011, 24027, 25714, 28143, 31188, 33792, 36137, 39203, 42920, 46294, 49108, 52580, 57165, 61365

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045849.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045849, A055406, A055413, A224212, A224213, A302862, A341400, A341401.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+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..44);  # Alois P. Heinz, Feb 10 2021

nmax = 44; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^7/(128 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^7 / (128 * (1 - x)).

a(n^2) = A055406(n).

A295849 Number of nonnegative solutions to gcd(x,y,z) = 1 and x^2 + y^2 + z^2 <= n.

Original entry on oeis.org

0, 3, 6, 7, 7, 13, 16, 16, 16, 19, 25, 28, 28, 34, 40, 40, 40, 49, 52, 55, 55, 61, 64, 64, 64, 70, 82, 85, 85, 97, 103, 103, 103, 109, 118, 124, 124, 130, 139, 139, 139, 154, 160, 163, 163, 169, 175, 175, 175, 181, 193, 199, 199, 211, 220, 220, 220, 226, 232, 241

Offset: 0

Seiichi Manyama, Nov 29 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000606, A048134, A295820, A295848.

N:= 100:
V:= Vector(N):
for x from 0 to floor(sqrt(N/3)) do
  for y from x to floor(sqrt((N-x^2)/2)) do
    for z from y to floor(sqrt(N-x^2-y^2)) do
      if igcd(x,y,z) = 1 then
        r:= x^2 + y^2 + z^2;
        m:= nops({x,y,z});
        if m=3 then V[r]:= V[r]+6
        elif m=2 then V[r]:= V[r]+3
        else V[r]:= V[r]+1
        fi
      fi
od od od:
0,op(ListTools:-PartialSums(convert(V,list))); # Robert Israel, Nov 30 2017

a[n_] := Sum[Boole[GCD[i, j, k] == 1], {i, 0, Sqrt[n]}, {j, 0, Sqrt[n - i^2]}, {k, 0, Sqrt[n - i^2 - j^2]}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

a(n) = {sum(i=0, sqrtint(n), sum(j=0, sqrtint(n-i^2), sum(k=0, sqrtint(n-i^2-j^2), gcd([i, j, k]) == 1)))} \\ Andrew Howroyd, Dec 12 2017

a(n) = a(n-1) + A295848(n) for n > 0.

A341403 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n.

Original entry on oeis.org

1, 9, 37, 93, 171, 283, 479, 767, 1076, 1420, 1952, 2688, 3444, 4228, 5320, 6776, 8262, 9662, 11454, 13918, 16480, 18832, 21772, 25644, 29508, 33044, 37300, 42732, 48340, 53556, 59632, 67472, 75405, 82237, 90189, 100661, 111155, 120403, 131099, 144651, 158469, 170621

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045850.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045850, A055407, A055414, A224212, A224213, A302862, A341400, A341401, A341402, A341404, A341405.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+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..41);  # Alois P. Heinz, Feb 10 2021

nmax = 41; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^8/(256 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^8 / (256 * (1 - x)).

a(n^2) = A055407(n).

cp's OEIS Frontend

A224212 Number of nonnegative solutions to x^2 + y^2 <= n.

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A002102 Number of nonnegative solutions to x^2 + y^2 + z^2 = n.

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A224213 Number of nonnegative solutions to x^2 + y^2 + z^2 + u^2 <= n.

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A000604 Number of nonnegative solutions to x^2 + y^2 + z^2 <= n^2.

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A302862 a(n) = [x^n] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

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A341400 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n.

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A341401 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n.

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A341402 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

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