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 1-6 of 6 results.

A302998 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] (1 + theta_3(x))^k/(2^k*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 11, 11, 5, 1, 1, 6, 20, 29, 17, 6, 1, 1, 7, 36, 70, 54, 26, 7, 1, 1, 8, 63, 157, 165, 99, 35, 8, 1, 1, 9, 106, 337, 482, 357, 163, 45, 9, 1, 1, 10, 171, 702, 1319, 1203, 688, 239, 58, 10, 1, 1, 11, 265, 1420, 3390, 3819, 2673, 1154, 344, 73, 11, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2018

Keywords

Comments

A(n,k) is the number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_k)^2 <= n^2.

Examples

			Square array begins:
  1,  1,   1,   1,    1,     1,  ...
  1,  2,   3,   4,    5,     6,  ...
  1,  3,   6,  11,   20,    36,  ...
  1,  4,  11,  29,   70,   157,  ...
  1,  5,  17,  54,  165,   482,  ...
  1,  6,  26,  99,  357,  1203,  ...

Links

Crossrefs

Columns k=0..10 give A000012, A000027, A000603, A000604, A055403, A055404, A055405, A055406, A055407, A055408, A055409.
Rows n=0..10 give A000012, A000027, A055417, A055418, A055419, A055420, A055421, A055422, A055423, A055424, A055425.
Main diagonal gives A302863.
Cf. A000122, A122510, A302996, A302997.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^k/(2^k (1 - x)), {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[x^i^2, {i, 0, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
  • PARI
    T(n,k)={if(k==0, 1, polcoef(((sum(j=0, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019

Formula

A(n,k) = [x^(n^2)] (1/(1 - x))*(Sum_{j>=0} x^(j^2))^k.

A000606 Number of nonnegative solutions to x^2 + y^2 + z^2 <= n.

Original entry on oeis.org

1, 4, 7, 8, 11, 17, 20, 20, 23, 29, 35, 38, 39, 45, 51, 51, 54, 63, 69, 72, 78, 84, 87, 87, 90, 99, 111, 115, 115, 127, 133, 133, 136, 142, 151, 157, 163, 169, 178, 178, 184, 199, 205, 208, 211, 223, 229, 229, 230, 239, 254, 260, 266, 278, 290, 290, 296
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • 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).

Links

Crossrefs

Cf. A000604, A117609.
Cf. A002102 (first differences).

Programs

  • Mathematica
    nn = 50; t = Table[0, {nn}]; Do[d = x^2 + y^2 + z^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}, {z, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)
  • Python
    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
            k += 1
  print(str(k), end=',')
# Alex Ratushnyak, Apr 01 2013

Formula

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^3. - Ilya Gutkovskiy, Mar 14 2017

Extensions

More terms from Sean A. Irvine, Dec 01 2010

A253663 Number of positive solutions to x^2+y^2+z^2 <= n^2.

Original entry on oeis.org

0, 0, 1, 7, 17, 38, 78, 127, 196, 296, 410, 564, 738, 958, 1220, 1514, 1848, 2235, 2686, 3175, 3719, 4365, 5007, 5758, 6568, 7442, 8415, 9477, 10597, 11779, 13100, 14459, 15954, 17566, 19231, 21029, 22916, 24930, 27030, 29293, 31616, 34103, 36732, 39459
Offset: 0

Views

Author

R. J. Mathar, Jan 07 2015

Keywords

Comments

Whereas A000604 counts solutions where x>=0, y>=0, z>=0, this sequence counts solutions where x>0, y>0, z>0.

Examples

			a(4)=17 counts the following solutions (x,y,z): (1,1,1), (2,2,2), three permutations of (1,1,2), three permutations of (1,1,3), three permutations of (1,2,2), and six permutations of (1,2,3).

Links

Crossrefs

Cf. A000604.

Programs

  • Sage
    [len([(x,y,z) for x in [1..n] for y in [1..n] for z in [1..n] if x^2+y^2+z^2<=n^2]) for n in [0..43]] # Tom Edgar, Jan 07 2015

Formula

a(n) = A211639(n^2).
a(n) = [x^(n^2)] (theta_3(x) - 1)^3/(8*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018
Comment from N. J. A. Sloane, Jun 02 2024 (Start)
The one-dimensional lattice {n: n an integer} , which graphically looks like
...o o o o o o ...
has theta series 1 + 2 q + 2 q^4 + 2 q^9 + 2 q^16 + ... = sum {n=-oo..oo} q^(n^2),
and that power series is called theta_3(q), A000122.
Raising it to the power 3 counts points with x^2+y^2+z^2 = k, A005875.
Dividing it by 1-x gives the partial sums, which basically is what this sequence is.
So a first approximation to a theta series for the sequence is theta_3(q)^8/(1-q).
Subtracting 1 and dividing by 8 is because here we only want positive solutions.
(End)

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}]

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.

A218711 Number of nonnegative solutions to x^2 + y^2 + z^2 < n^2.

Original entry on oeis.org

0, 1, 8, 23, 51, 90, 157, 230, 341, 471, 639, 835, 1063, 1340, 1671, 2022, 2443, 2893, 3428, 4004, 4653, 5359, 6133, 6977, 7907, 8886, 9991, 11152, 12428, 13724, 15192, 16683, 18358, 20072, 21932, 23880, 25941, 28117, 30397, 32822, 35376, 38013, 40840, 43765, 46880, 50090, 53448, 56911, 60583, 64379
Offset: 0

Views

Author

Jon Perry, Nov 04 2012

Keywords

Crossrefs

Cf. A000604, A181788.

Programs

  • JavaScript
    for (i=0;i<50;i++) {
d=0;
for (a=0;a<=i;a++)
for (b=0;b<=i;b++)
for (c=0;c<=i;c++)
if (Math.pow(a,2)+Math.pow(b,2)+Math.pow(c,2)

Formula

a(n) = A000604(n) - A181788(n).
Showing 1-6 of 6 results.

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

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