cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A302995 a(n) = [x^(n^2)] (theta_3(x) - 1)^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 1, 7, 32, 177, 1269, 9263, 74452, 652710, 6078048, 60447082, 631870024, 6915613084, 79113376037, 941759419159, 11630647314564, 148799595377384, 1966441829785081, 26793749867965515, 375812005722920406, 5416574818546042067, 80123280319100908258, 1214860029446181979357
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2018

Keywords

Comments

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

Links

Crossrefs

Cf. A000122, A001182, A253663, A298330, A302861, A302863.

Programs

  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^n/(2^n (1 - x)), {x, 0, n^2}], {n, 0, 23}]
Join[{1}, Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 1, n}]^n, {x, 0, n^2}], {n, 23}]]

A341423 Number of positive solutions to (x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 <= n^2.

Original entry on oeis.org

1, 5, 32, 94, 219, 437, 804, 1362, 2177, 3271, 4768, 6708, 9227, 12381, 16254, 20954, 26707, 33461, 41480, 50884, 61703, 74183, 88606, 104862, 123481, 144241, 167604, 193648, 222799, 254731, 290244, 329512, 372545, 419661, 470822, 526646, 587481, 653505
Offset: 2

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A046895, A055403, A055410, A063730, A224213, A253663, A302995, A341424, A341425, A341426, A341427, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 4):
seq(a(n), n=2..39);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^4/(16 (1 - x)), {x, 0, n^2}], {n, 2, 39}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^4 / (16 * (1 - x)).

A341424 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n^2.

Original entry on oeis.org

6, 51, 177, 547, 1348, 2958, 5574, 10084, 16974, 27450, 41970, 62671, 90216, 128082, 175867, 238018, 316373, 414998, 534094, 682144, 859705, 1075165, 1326551, 1627896, 1976582, 2390057, 2862607, 3411273, 4039483, 4760419, 5571729, 6500650, 7541560, 8722096
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055404, A055411, A175360, A253663, A302995, A340481, A341400, A341423, A341425, A341426, A341427, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 5):
seq(a(n), n=3..36);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^5/(32 (1 - x)), {x, 0, n^2}], {n, 3, 36}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^5 / (32 * (1 - x)).

A341425 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n^2.

Original entry on oeis.org

7, 48, 331, 1269, 3698, 9382, 20927, 42683, 79844, 142173, 238810, 387615, 603589, 915324, 1345294, 1939221, 2729723, 3783313, 5138567, 6895632, 9108626, 11909496, 15362753, 19642539, 24832744, 31179476, 38757032, 47877886, 58647957, 71447776, 86391220
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055405, A055412, A175361, A253663, A302995, A340905, A341401, A341423, A341424, A341426, A341427, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 6):
seq(a(n), n=3..33);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^6/(64 (1 - x)), {x, 0, n^2}], {n, 3, 33}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^6 / (64 * (1 - x)).

A341426 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n^2.

Original entry on oeis.org

1, 71, 491, 2522, 9263, 27723, 71480, 163908, 345657, 679802, 1252185, 2203724, 3715206, 6041979, 9510283, 14591324, 21788606, 31894205, 45741815, 64467383, 89363919, 122254946, 164721244, 219526449, 289133792, 377013829, 486522424, 622759365
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055406, A055413, A253663, A302995, A340906, A341396, A341402, A341423, A341424, A341425, A341427, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 7):
seq(a(n), n=3..30);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^7/(128 (1 - x)), {x, 0, n^2}], {n, 3, 30}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^7 / (128 * (1 - x)).

A341427 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n^2.

Original entry on oeis.org

1, 45, 767, 4452, 21178, 74452, 224313, 586035, 1387583, 2999430, 6102276, 11656386, 21282969, 37159993, 62687904, 102213426, 162345824, 251064745, 379922217, 562833191, 819351646, 1171991382, 1651937498, 2294227971, 3147090871, 4263499419
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055407, A055414, A253663, A302995, A340915, A341397, A341403, A341423, A341424, A341425, A341426, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 8):
seq(a(n), n=3..28);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^8/(256 (1 - x)), {x, 0, n^2}], {n, 3, 28}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^8 / (256 * (1 - x)).

A341428 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n^2.

Original entry on oeis.org

1, 46, 760, 7751, 43910, 186098, 652710, 1943742, 5178030, 12411211, 27773308, 57798904, 114152429, 214399664, 387571706, 673189698, 1135916808, 1857320784, 2966816950, 4623984661, 7066527283, 10577150039, 15589368584, 22580091614, 32256768126
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055408, A055415, A253663, A302995, A340946, A341398, A341404, A341423, A341424, A341425, A341426, A341427, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 9):
seq(a(n), n=3..27);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^9/(512 (1 - x)), {x, 0, n^2}], {n, 3, 27}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^9 / (512 * (1 - x)).

A341429 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n^2.

Original entry on oeis.org

56, 1108, 12098, 84624, 439371, 1785368, 6078048, 18139393, 48586117, 118929400, 270250734, 578320470, 1169522013, 2261784392, 4193751331, 7509793133, 13008356489, 21921125415, 35951569269, 57666975238, 90464266824, 139295784464, 210514511189, 313228848537
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055409, A055416, A253663, A302995, A340947, A341399, A341405, A341423, A341424, A341425, A341426, A341427, A341428.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 10):
seq(a(n), n=4..27);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^10/(1024 (1 - x)), {x, 0, n^2}], {n, 4, 27}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^10 / (1024 * (1 - x)).

A349610 Number of solutions to x^2 + y^2 + z^2 <= n^2, where x, y, z are positive odd integers.

Original entry on oeis.org

0, 0, 1, 1, 4, 7, 17, 20, 35, 45, 69, 84, 114, 136, 184, 217, 272, 314, 389, 443, 528, 597, 702, 784, 901, 1018, 1166, 1268, 1442, 1589, 1791, 1926, 2157, 2332, 2584, 2800, 3058, 3293, 3596, 3872, 4194, 4485, 4878, 5184, 5590, 5950, 6388, 6761, 7232
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 23 2021

Keywords

Examples

			a(4) = 4 since there are solutions (1,1,1), (3,1,1), (1,3,1), (1,1,3).

Links

Crossrefs

Cf. A000604, A000605, A008437, A053596, A085121, A253663, A349609, A349611.

Programs

  • Mathematica
    Table[SeriesCoefficient[EllipticTheta[2, 0, x^4]^3/(8 (1 - x)), {x, 0, n^2}], {n, 0, 48}]

Formula

a(n) = [x^(n^2)] theta_2(x^4)^3 / (8 * (1 - x)).
a(n) = Sum_{k=0..n^2} A008437(k).
a(n) = A053596(n) / 8.

A212095 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>=x^2+y^2+z^2.

Original entry on oeis.org

0, 0, 1, 8, 25, 63, 141, 268, 464, 760, 1170, 1734, 2472, 3430, 4650, 6164, 8012, 10247, 12933, 16108, 19827, 24192, 29199, 34957, 41525, 48967, 57382, 66859, 77456, 89235, 102335, 116794, 132748, 150314, 169545, 190574, 213490, 238420
Offset: 0

Views

Author

Clark Kimberling, May 02 2012

Keywords

Comments

a(n)+A212092(n)=n^4. For a guide to related sequences, see A211795.

Crossrefs

Cf. A211795.
Partial sums of A253663.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 >= x^2 + y^2 + z^2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212095 *)
(* Peter J. C. Moses, Apr 13 2012 *)
Showing 1-10 of 10 results.

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