A302998 -id:A302998 - cp's OEIS frontend

A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 33, 29, 9, 1, 1, 11, 89, 123, 49, 11, 1, 1, 13, 221, 425, 257, 81, 13, 1, 1, 15, 485, 1343, 1281, 515, 113, 15, 1, 1, 17, 953, 4197, 5913, 3121, 925, 149, 17, 1, 1, 19, 1713, 12435, 23793, 16875, 6577, 1419, 197, 19, 1, 1, 21, 2869, 33809, 88273, 84769, 42205, 11833, 2109, 253, 21, 1

Offset: 0

Ilya Gutkovskiy, Apr 17 2018

A(n,k) is the number of integer lattice points inside the k-dimensional hypersphere of radius n.

			Square array begins:
  1,   1,   1,    1,     1,      1,  ...
  1,   3,   5,    7,     9,     11,  ...
  1,   5,  13,   33,    89,    221,  ...
  1,   7,  29,  123,   425,   1343,  ...
  1,   9,  49,  257,  1281,   5913,  ...
  1,  11,  81,  515,  3121,  16875,  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Columns k=0..10 give A000012, A005408, A000328, A000605, A055410, A055411, A055412, A055413, A055414, A055415, A055416.
Rows k=0..10 give A000012, A005408, A055426, A055427, A055428, A055429, A055430, A055431, A055432, A055433, A055434.
Main diagonal gives A302861.
Cf. A000122, A122510, A302996, A302998.

Table[Function[k, SeriesCoefficient[EllipticTheta[3, 0, x]^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, -n, n}]^k, {x, 0, n^2}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

T(n,k)={if(k==0, 1, polcoef(((1 + 2*sum(j=1, n, x^(j^2)) + O(x*x^(n^2)))^k)/(1-x), n^2))} \\ Andrew Howroyd, Sep 14 2019

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

A000603 Number of nonnegative solutions to x^2 + y^2 <= n^2.

Original entry on oeis.org

1, 3, 6, 11, 17, 26, 35, 45, 58, 73, 90, 106, 123, 146, 168, 193, 216, 243, 271, 302, 335, 365, 402, 437, 473, 516, 557, 600, 642, 687, 736, 782, 835, 886, 941, 999, 1050, 1111, 1167, 1234, 1297, 1357, 1424, 1491, 1564, 1636, 1703, 1778, 1852, 1931, 2012, 2095

Offset: 0

N. J. A. Sloane

nonn

Row sums of triangle A255238. - Wolfdieter Lang, Mar 15 2015

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..1000

Column k=2 of A302998.

Cf. A000328, A036695, A036702, A255238, A255195.

a000603 n = length [(x,y) | x <- [0..n], y <- [0..n], x^2 + y^2 <= n^2]
-- Reinhard Zumkeller, Jan 23 2012

Table[cnt = 0; Do[If[x^2 + y^2 <= n^2, cnt++], {x, 0, n}, {y, 0, n}]; cnt, {n, 0, 51}] (* T. D. Noe, Apr 02 2013 *)
Table[If[n==1,1,2*Sum[Sum[A255195[[n, n - k + 1]], {k, 1, k}], {k, 1, n}] - Ceiling[(n - 1)/Sqrt[2]]],{n,1,52}] (* Mats Granvik, Feb 19 2015 *)

a(n)=my(n2=n^2);sum(a=0,n,sqrtint(n2-a^2)+1) \\ Charles R Greathouse IV, Apr 03 2013

from math import isqrt
def A000603(n): return (m:=n<<1)+sum(isqrt(k*(m-k)) for k in range(1,n))+1 # Chai Wah Wu, Jul 18 2024

a(n) = n^2 * Pi/4 + O(n). - Charles R Greathouse IV, Apr 03 2013
a(n) = A001182(n) + 2*n + 1. - R. J. Mathar, Jan 07 2015
a(n) = 2*A026702(n) - (1 + floor(n/sqrt(2))), n >= 0. - Wolfdieter Lang, Mar 15 2015
a(n) = [x^(n^2)] (1 + theta_3(x))^2/(4*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

More terms from David W. Wilson, May 22 2000

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

A055406 Number of points in N^7 of norm <= n.

Original entry on oeis.org

1, 8, 106, 702, 3390, 11496, 33792, 83877, 191433, 394251, 766518, 1397065, 2446316, 4076126, 6601783, 10324698, 15776495, 23436651, 34198045, 48828417, 68680634, 94868383, 129404333, 173984393, 231540046, 304058667, 395837165

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Column k=7 of A302998.

a[n_]:=SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^7/(128 (1 - x)), {x, 0, n^2}]; Array[a,27,0] (* Stefano Spezia, Mar 31 2025 *)

a(n) = [x^(n^2)] (1 + theta_3(x))^7/(128*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

A055407 Number of points in N^8 of norm <= n.

Original entry on oeis.org

1, 9, 171, 1420, 8262, 33044, 111155, 312913, 793049, 1807754, 3834490, 7601982, 14333819, 25683401, 44367189, 73913316, 119544010, 187884853, 288597817, 433331391, 638511294, 923461382, 1314459790, 1842974369, 2550288371

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Column k=8 of A302998.

a(n) = [x^(n^2)] (1 + theta_3(x))^8/(256*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

A055408 Number of points in N^9 of norm <= n.

Original entry on oeis.org

1, 10, 265, 2780, 19250, 91067, 349122, 1115334, 3134307, 7903501, 18286018, 39418159, 79946582, 154050794, 283766861, 503208408, 861586362, 1432132554, 2314882321, 3655071196, 5640732550, 8538547541, 12683700617, 18541085573

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Column k=9 of A302998.

a(n) = [x^(n^2)] (1 + theta_3(x))^9/(512*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

A055409 Number of points in N^10 of norm <= n.

Original entry on oeis.org

1, 11, 396, 5258, 43154, 241283, 1053323, 3813001, 11887373, 33107532, 83570177, 195642892, 427116918, 883782766, 1736506732, 3275680338, 5938894018, 10431199387, 17749369808, 29449543675, 47615028191, 75399727379

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Column k=10 of A302998.

a(n) = [x^(n^2)] (1 + theta_3(x))^10/(1024*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

A055403 Number of points in N^4 of norm <= n.

Original entry on oeis.org

1, 5, 20, 70, 165, 357, 688, 1154, 1867, 2907, 4272, 6070, 8357, 11307, 14998, 19470, 24809, 31211, 38928, 47816, 58201, 70361, 83962, 99648, 117519, 137521, 160270, 185528, 213615, 244649, 279370, 317006, 358809, 404823, 454838, 509486, 568493

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Column k=4 of A302998.

a(n) = [x^(n^2)] (1 + theta_3(x))^4/(16*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

A055404 Number of points in N^5 of norm <= n.

Original entry on oeis.org

1, 6, 36, 157, 482, 1203, 2673, 5139, 9389, 15930, 25810, 39855, 59711, 86346, 122467, 168957, 229743, 305153, 400844, 518154, 662629, 835490, 1044410, 1292216, 1588236, 1930057, 2334622, 2798143, 3340038, 3955063, 4663169

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Column k=5 of A302998.

a(n) = [x^(n^2)] (1 + theta_3(x))^5/(32*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

A055405 Number of points in N^6 of norm <= n.

Original entry on oeis.org

1, 7, 63, 337, 1319, 3819, 9763, 21374, 43774, 81586, 145138, 243345, 394696, 612274, 928616, 1363401, 1967428, 2762729, 3825273, 5197436, 6974412, 9198648, 12019669, 15500750, 19827441, 25050955, 31438147, 39053729, 48254762

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Column k=6 of A302998.

a(n) = [x^(n^2)] (1 + theta_3(x))^6/(64*(1 - x)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 15 2018

cp's OEIS Frontend

A302997 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = [x^(n^2)] theta_3(x)^k/(1 - x), where theta_3() is the Jacobi theta function.

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A000603 Number of nonnegative solutions to x^2 + y^2 <= n^2.

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

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A055406 Number of points in N^7 of norm <= n.

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A055407 Number of points in N^8 of norm <= n.

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A055408 Number of points in N^9 of norm <= n.

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A055409 Number of points in N^10 of norm <= n.

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A055403 Number of points in N^4 of norm <= n.

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A055404 Number of points in N^5 of norm <= n.

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