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 1-7 of 7 results.

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A038671.

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A045848.

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A045849.

Links

Crossrefs

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

Programs

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

Formula

G.f.: (1 + theta_3(x))^7 / (128 * (1 - x)).
a(n^2) = A055406(n).

A302863 a(n) = [x^(n^2)] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 6, 29, 165, 1203, 9763, 83877, 793049, 7903501, 83570177, 933697153, 10905583809, 133352809334, 1695473999478, 22354920990148, 305096197935075, 4296142551821184, 62336908825014452, 930284705538262688, 14255992611680074754, 224065160215526683317, 3607018540134004189466
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 14 2018

Keywords

Comments

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

Links

Crossrefs

Main diagonal of A302998.
Cf. A000603, A000604, A010052, A055403, A055404, A055405, A055406, A055407, A055408, A055409, A298329, A302861, A302862.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n^2}], {n, 0, 22}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n^2}], {n, 0, 22}]

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A045850.

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A045851.

Links

Crossrefs

Cf. A000122, A000606, A003059, A045851, A055408, A055415, A224212, A224213, A302862, A341400, A341401, A341402, A341403, A341405.

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Comments

Partial sums of A045852.

Links

Crossrefs

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

Programs

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

Formula

G.f.: (1 + theta_3(x))^10 / (1024 * (1 - x)).
a(n^2) = A055409(n).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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