A224212 -id:A224212 - cp's OEIS frontend

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

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

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).

A295820 Number of nonnegative solutions to (x,y) = 1 and x^2 + y^2 <= n.

Original entry on oeis.org

0, 2, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 13, 15, 15, 15, 17, 17, 17, 17, 17, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 23, 23, 25, 25, 25, 27, 27, 27, 27, 27, 29, 29, 29, 31, 31, 31, 31, 35, 35, 35, 35, 35, 35

Offset: 0

Seiichi Manyama, Nov 28 2017

nonn

			Solutions to (x,y) = 1 and x^2 + y^2 <= 17;
  *         (1,4)
  * *       (1,3), (2,3)
  *   *     (1,2), (3,2)
* * * * *   (0,1), (1,1), (2,1), (3,1), (4,1)
  *         (1,0)
            a(17) = 11.

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

Cf. A049643, A224212, A295819, A295849.

a[n_] := Sum[Boole[GCD[i, j]==1 ], {i, 0, Sqrt[n]}, {j, 0, Sqrt[n-i^2]}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)

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

a(n) = a(n-1) + A295819(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).

A341404 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

Original entry on oeis.org

1, 10, 46, 130, 265, 463, 799, 1339, 2014, 2780, 3860, 5444, 7301, 9263, 11783, 15263, 19250, 23237, 27893, 34193, 41519, 48701, 56765, 67421, 79484, 91067, 103739, 119855, 138035, 155819, 174923, 198863, 225890, 251444, 277976, 311492, 349122, 384420, 421284

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045851.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045851, A055408, A055415, A224212, A224213, A302862, A341400, A341401, A341402, A341403, 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, 9)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..38);  # Alois P. Heinz, Feb 10 2021

nmax = 38; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^9/(512 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^9 / (512 * (1 - x)).

a(n^2) = A055408(n).

A341405 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

Original entry on oeis.org

1, 11, 56, 176, 396, 738, 1308, 2268, 3618, 5258, 7449, 10689, 14889, 19609, 25369, 33289, 43154, 53774, 65739, 81339, 100671, 121221, 143421, 171501, 205701, 241283, 278678, 324398, 378998, 435968, 495428, 566468, 650798, 737888, 826083, 930123, 1053323

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045852.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045852, A055409, A055416, A224212, A224213, A302862, A341400, A341401, A341402, A341403, A341404.

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, 10)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 10 2021

nmax = 36; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^10/(1024 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^10 / (1024 * (1 - x)).

a(n^2) = A055409(n).

A372511 Number of solutions to x^2 + y^2 <= n, where x, y are positive odd integers.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17

Offset: 0

Ilya Gutkovskiy, May 04 2024

nonn

Index entries for sequences related to sums of squares

Cf. A008442, A057655, A224212, A237526, A290081, A349609, A372512.

nmax = 85; CoefficientList[Series[EllipticTheta[2, 0, x^4]^2/(4 (1 - x)), {x, 0, nmax}], x]

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A295820 Number of nonnegative solutions to (x,y) = 1 and x^2 + y^2 <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A341404 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

Views

Author

Keywords

Comments

Links