A036702 -id:A036702 - cp's OEIS frontend

A192710 T(i,j,k) = Number of i X j integer matrices with each row summing to zero, row elements in nondecreasing order, rows in lexicographically nondecreasing order, and the sum of squares of the elements <= 2k^2 (number of sets of i zero-sum j-vectors with total modulus squared not more than 2k^2, ignoring vector and component permutations), 3d array by constant coordinate sum planes: (((T(i+1,j+1,s-i-j+1), j=0..s-i), i=0..s), s=0..infinity).

1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4, 5, 2, 1, 4, 2, 1, 2, 1, 1, 5, 8, 6, 2, 1, 7, 8, 2, 1, 5, 2, 1, 2, 1, 1, 6, 13, 15, 8, 2, 1, 10, 20, 11, 2, 1, 10, 9, 2, 1, 6, 2, 1, 2, 1, 1, 7, 18, 26, 21, 9, 2, 1, 15, 54, 48, 13, 2, 1, 16, 36, 13, 2, 1, 12, 10, 2, 1, 6, 2, 1, 2, 1, 1, 8, 25, 45, 48, 28, 9, 2, 1, 20, 104

Offset: 1

R. H. Hardin Jul 07 2011

			Some solutions for n=245, T(3,4,6)
.-3..1..1..1...-5..0..0..5...-5..0..2..3...-4..1..1..2...-5.-1..3..3
.-2.-2..1..3...-2.-1..1..2...-3.-1..2..2...-2.-2..0..4...-1.-1.-1..3
.-2.-1.-1..4...-2..0..1..1...-2.-2..2..2...-2.-1..1..2...-1.-1.-1..3

R. H. Hardin, Table of n, a(n) for n = 1..1640

Column T(1,3,n) is A000982(n+1).

Column T(2,2,n) is A036702(n).

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

A036700 Number of Gaussian integers z=a+bi satisfying |z|<=n, a>=0, 0<=b

Original entry on oeis.org

0, 1, 2, 4, 7, 11, 15, 20, 26, 33, 41, 49, 57, 68, 79, 91, 102, 115, 129, 144, 160, 175, 193, 210, 228, 249, 269, 290, 311, 333, 357, 380, 406, 431, 458, 487, 512, 542, 570, 603, 634, 664, 697, 730, 766, 802, 835, 872, 909, 948, 988

Offset: 0

Clark Kimberling

nonn

Wikipedia, Gaussian Integers
Index entries for Gaussian integers and primes

Cf. A036701, A036702.

A036700 := proc(n)
        local a,x,y ;
        a := 0 ;
        for x from 0 do
                if x^2 > n^2 then
                        return a;
                fi ;
                for y from 0 to x-1 do
                        if y^2+x^2 <= n^2 then
                                a := a+1 ;
                        end if;
                end do;
        end do:
end proc: # R. J. Mathar, Oct 29 2011

Partial sums of A036701. - Sean A. Irvine, Nov 22 2020

A255250 Array T(n, m) of numbers of points of a square lattice in the first octant covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 1, 5, 3, 2, 6, 4, 3, 2, 7, 5, 4, 3, 1, 8, 6, 5, 4, 2, 9, 7, 6, 5, 3, 2, 10, 8, 7, 6, 5, 3, 1, 11, 9, 8, 7, 6, 4, 3, 1, 12, 10, 9, 8, 7, 5, 4, 2, 13, 11, 10, 9, 8, 6, 5, 3, 1, 14, 12, 11, 10, 9, 8, 6, 4, 3, 1, 15, 13, 12, 11, 10, 9, 7, 6, 4, 2, 16, 14, 13, 12, 11, 10, 8, 7, 5, 4, 2

Offset: 0

Wolfdieter Lang, Mar 14 2015

The row length of this array (irregular triangle) is 1 + flpoor(n/sqrt(2)) = 1 + A049472(n) = 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, ...
This entry is motivated by the proposal A255195 by Mats Granvik, who gave the first differences of this array.
See the MathWorld link on Gauss's circle problem.
The first octant of a square lattice (x, y) with n = x >= y = m >= 0 is considered. The number of lattice points in this octant covered by a circular disk of radius R = n around the origin having ordinate value y = m are denoted by T(n, m), for n >= 0 and m = 0, 1, ..., floor(n/sqrt(2)).
The row sums give RS(n) = A036702(n), n >= 0. This is the total number of square lattice points in the first octant covered by a circular disk of radius R = n.
The alternating row sums give A256094(n), n >= 0.
The total number of square lattice points in the first quadrant covered by a circular disk of radius R = n is therefore 2*RS(n) - (1 + floor(n/sqrt(2))) = A000603(n).

			The array (irregular triangle) T(n, m) begins:
n\m  0  1  2  3  4  5 6 7 8 9 10 ....
0:   1
1:   2
2:   3  1
3:   4  2  1
4:   5  3  2
5:   6  4  3  2
6:   7  5  4  3  1
7:   8  6  5  4  2
8:   9  7  6  5  3  2
9:  10  8  7  6  5  3 1
10: 11  9  8  7  6  4 3 1
11: 12 10  9  8  7  5 4 2
12: 13 11 10  9  8  6 5 3 1
13: 14 12 11 10  9  8 6 4 3 1
14: 15 13 12 11 10  9 7 6 4 2
15: 16 14 13 12 11 10 8 7 5 4  2
...

E. W. Weisstein, World of Mathematics, Gauss's Circle Problem.

Cf. A036702, A256094, A000328, A049472, A255195, A255238 (quadrant).

T(n, m) = floor(sqrt(n^2 - m^2)) - (m-1), n >= 0, m = 0, 1, ..., floor(n/sqrt(2)).

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

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A036700 Number of Gaussian integers z=a+bi satisfying |z|<=n, a>=0, 0<=b

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A255250 Array T(n, m) of numbers of points of a square lattice in the first octant covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m.

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