A001182 -id:A001182 - cp's OEIS frontend

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

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

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

Ilya Gutkovskiy, Apr 17 2018

nonn

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

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

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

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

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Index entries for sequences related to sums of squares

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

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

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^4/(16 (1 - x)), {x, 0, n^2}], {n, 2, 39}]

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

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Index entries for sequences related to sums of squares

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

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

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^5/(32 (1 - x)), {x, 0, n^2}], {n, 3, 36}]

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

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Index entries for sequences related to sums of squares

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

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

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^6/(64 (1 - x)), {x, 0, n^2}], {n, 3, 33}]

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

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Index entries for sequences related to sums of squares

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

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

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^7/(128 (1 - x)), {x, 0, n^2}], {n, 3, 30}]

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

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Index entries for sequences related to sums of squares

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

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

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^8/(256 (1 - x)), {x, 0, n^2}], {n, 3, 28}]

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

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Index entries for sequences related to sums of squares

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

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

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^9/(512 (1 - x)), {x, 0, n^2}], {n, 3, 27}]

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

Ilya Gutkovskiy, Feb 11 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Index entries for sequences related to sums of squares

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

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

Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^10/(1024 (1 - x)), {x, 0, n^2}], {n, 4, 27}]

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

A125228 Maximal number of squares of side 1 in a disk of radius n.

Original entry on oeis.org

1, 7, 21, 39, 65, 93, 135, 179, 227, 285, 349, 415, 495, 573, 663, 759, 859, 963, 1071, 1199, 1325, 1457, 1591, 1735, 1891, 2049, 2207, 2383, 2557, 2735, 2929, 3123, 3327, 3529, 3739, 3955, 4191, 4427, 4665, 4901, 5159, 5413, 5681, 5951, 6231, 6515, 6799

Offset: 1

Filippo ALUFFI PENTINI (falpen(AT)gmail.com), Jan 25 2007

			a(2)=7 since you cannot pack more than 7 unit-side squares in a disk of radius 2

David Dewan, Table of n, a(n) for n = 1..10000
David Dewan, Drawings for n=1..12

Similar to A001182 but less constrained.

A124484 is another version.

f[n_] := 2 Sum[ IntegerPart[2 Sqrt[n^2 - (n - k - 1/2)^2]], {k, 0, n - 2}] + IntegerPart[2 Sqrt[n^2 - 1/2^2]]; Array[f, 47] (* Robert G. Wilson v, Jan 27 2007 *)
a[n_]:=2 Sum[Floor[2 Sqrt[n^2-(k+1/2)^2]],{k,n-1}]+2n-1;
Array[a, 47]  (*  David Dewan, Jun 07 2024*)

from math import isqrt
def A125228(n): return (m:=n<<1)-1+(sum(isqrt((k*(m-k+1)-n<<2)-1) for k in range(1,n))<<1) # Chai Wah Wu, Jul 18 2024

a(n) = 2*Sum_{k=1..n-1} floor(2*sqrt(n^2 - (k+1/2)^2)) + 2*n - 1.

More terms from Robert G. Wilson v, Jan 27 2007

cp's OEIS Frontend

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

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

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

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

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

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

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

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

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