A000328 -id:A000328 - cp's OEIS frontend

A053411 Circle numbers (version 1): a(n)= number of points (i,j), i,j integers, contained in a circle of diameter n, centered at the origin.

1, 1, 5, 9, 13, 21, 29, 37, 49, 69, 81, 97, 113, 137, 149, 177, 197, 225, 253, 293, 317, 349, 377, 421, 441, 489, 529, 577, 613, 665, 709, 749, 797, 861, 901, 973, 1009, 1085, 1129, 1201, 1257, 1313, 1373, 1457, 1517, 1597, 1653, 1741, 1793, 1885, 1961

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 10 2000

a(n)/(n/2)^2 -> Pi.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A053414, A053415, A053416, A053417.

Bisections: A000328 and A036704.

a[n_] := (m = Ceiling[n/2]; Sum[Boole[i^2 + j^2 <= n^2/4], {i, -m, m}, {j, -Ceiling @ Sqrt[ m^2 - i^2 ], Ceiling @ Sqrt[ m^2 - i^2 ]}]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 06 2013 *)

A049735 Array T(i,j) is the number of lattice points (x,y) in circle with radius (0,0)-to-(i,j), read by antidiagonals.

Original entry on oeis.org

1, 5, 5, 13, 9, 13, 29, 21, 21, 29, 49, 37, 25, 37, 49, 81, 57, 45, 45, 57, 81, 113, 89, 69, 61, 69, 89, 113, 149, 121, 97, 81, 81, 97, 121, 149, 197, 161, 129, 109, 101, 109, 129, 161, 197, 253, 213, 177, 145, 137, 137, 145, 177, 213, 253

Offset: 0

Clark Kimberling

Specifically, x^2 + y^2 <= i^2 + j^2.

			Antidiagonals (each starting on row 0):
  {1},
  {5, 5},
  {13, 9, 13},
  ...
Array begins:
   1  5 13  29  49  81
   5  9 21  37  57  89
  13 21 25  45  69  97
  29 37 45  61  81 109
  49 57 69  81 101 137
  81 89 97 109 137 161

Cf. A000328 (1st column or row).

T(n, k) = my(z=norml2([n, k]), m=ceil(sqrt(2)*max(n,k))); sum(x=-m, m, sum(y=-m, m, norml2([x, y]) <= z)); \\ Michel Marcus, Aug 07 2021

T(n,0) = A000328(n).

A247588 Number of integer-sided acute triangles with largest side n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, 52, 56, 63, 67, 73, 80, 84, 90, 96, 104, 111, 117, 126, 132, 140, 147, 154, 165, 172, 183, 189, 198, 210, 219, 229, 237, 247, 260, 270, 282, 292, 302

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(3) = 3 because there are 3 integer-sided acute triangles with largest side 3: (1,3,3); (2,3,3); (3,3,3).

Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A046080, A002623, A224921, A247586, A247587, A247589.

tr_a:=proc(n) local a,b,t,d;t:=0:
for a to n do
for b from max(a,n+1-a) to n do
d:=a^2+b^2-n^2:
if d>0 then t:=t+1 fi
od od;
t; end;

a[ n_] := Length @ FindInstance[ n >= b >= a >= 1 && n < b + a && n^2 < b^2 + a^2, {a, b}, Integers, 10^9]; (* Michael Somos, May 24 2015 *)

a(n) = sum(j=0, n*(1 - sqrt(2)/2), n - j - floor(sqrt(2*j*n - j^2))); \\ Michel Marcus, Oct 07 2014

{a(n) = sum(j=0, n - sqrtint(n*n\2) - 1, n - j - sqrtint(2*j*n - j*j))}; /* Michael Somos, May 24 2015 */

a(n) = Sum_{j=0..floor(n*(1 - sqrt(2)/2))} (n - j - floor(sqrt(2*j*n - j^2))). - Anton Nikonov, Oct 06 2014

a(n) = (1/8)*(-4*ceiling((n - 1)/sqrt(2)) + 4*n^2 - A000328(n) + 1), n > 1. - Mats Granvik, May 23 2015

A046112 a(n) is smallest integral radius of circle centered at (0,0) having 8n-4 lattice points on its circumference; a(n)/2 is smallest half-integral radius circle centered at (1/2,0) having 4n-2 lattice points; a(n)/3 is smallest third-integral radius circle centered at (1/3,0) having 2n-1 lattice points.

Original entry on oeis.org

1, 5, 25, 125, 65, 3125, 15625, 325, 390625, 1953125, 1625, 48828125, 4225, 1105, 6103515625, 30517578125, 40625, 21125, 3814697265625, 203125, 95367431640625, 476837158203125, 5525, 11920928955078125, 274625

Offset: 1

Eric W. Weisstein

nonn

Ray Chandler, Table of n, a(n) for n = 1..1439 (a(1440) exceeds 1000 digits).
Eric Weisstein's World of Mathematics, Circle Lattice Points
Eric Weisstein's World of Mathematics, Pythagorean Triple

Except for offset, same as A006339.

Cf. A046109, A046110, A046111, A000328.

A036702 a(n)=number of Gaussian integers z=a+bi satisfying |z|<=n, a>=0, 0<=b<=a.

Original entry on oeis.org

1, 2, 4, 7, 10, 15, 20, 25, 32, 40, 49, 57, 66, 78, 89, 102, 114, 128, 142, 158, 175, 190, 209, 227, 245, 267, 288, 310, 331, 354, 379, 402, 429, 455, 483, 512, 538, 569, 597, 631, 663, 693, 727, 761, 798, 834, 868, 906, 943, 983

Offset: 0

Clark Kimberling

Row sums of the irregular triangle A255250. - Wolfdieter Lang, Mar 15 2015

Index entries for Gaussian integers and primes

Cf. A036700, A049472, A000603, A000328, A255250.

A036702 := 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 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

a[n_] := Module[{a, b}, If[n == 0, 1, Reduce[a^2 + b^2 <= n^2 && a >= 0 && 0 <= b <= a, {a, b}, Integers] // Length]];
a /@ Range[0, 49] (* Jean-François Alcover, Oct 17 2019 *)

a(n) - A036700(n) = 1+A049472(n). - R. J. Mathar, Oct 29 2011

a(n) = sum(floor(sqrt(n^2 - m^2)) - (m-1), m = 0.. floor(n/sqrt(2))), n >= 0. See A255250. - Wolfdieter Lang, Mar 15 2015

A062711 Number of prime Gaussian integers z=a+bi with |z|<=n.

Original entry on oeis.org

0, 1, 4, 6, 8, 10, 15, 19, 21, 25, 32, 34, 38, 44, 46, 52, 60, 66, 73, 79, 87, 93, 98, 104, 114, 122, 128, 138, 146, 154, 163, 173, 181, 193, 203, 213, 221, 231, 239, 245, 259, 273, 280, 294, 304, 316, 327, 343, 359, 369

Offset: 1

Reiner Martin, Jul 14 2001

Cf. A000328, A062327, A000720.

m = 50;
t = Table[x + y I, {x, -m, m}, {y, -m, m}] // Flatten[#, 1]& // Select[#, PrimeQ[#, GaussianIntegers -> True]& ]& // Sort // DeleteDuplicates[#, Abs[#1] == Abs[#2] && MatchQ[#1 /#2 , 1|-1|I|-I]& ]&;
a[n_] := Select[t, Abs[#] <= n&] // Length;
Array[a, m] (* Jean-François Alcover, Jul 29 2016 *)

Two prime Gaussian integers are not counted separately if they are associated, i.e. if their quotient is a unit (1, i, -1 or -i).
Similar to the ordinary prime number theorem (see A000720) we have the asymptotic expression: a(n) ~ n^2/(2 * log(n)) - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 16 2001
a(1)=0, a(n)=1+A066339(n^2)+A066490(n) for n>0. - T. D. Noe, Feb 20 2007

A077770 Number of ordered pairs of integers (x,y) with n^2 < x^2 + y^2 < (n+1)^2; number of lattice points between circles of radii n and n+1.

Original entry on oeis.org

0, 4, 12, 16, 20, 28, 32, 44, 52, 52, 56, 60, 76, 80, 84, 84, 92, 104, 116, 116, 112, 140, 132, 136, 148, 148, 164, 160, 164, 180, 176, 204, 196, 204, 216, 196, 228, 216, 252, 236, 224, 260, 260, 284, 272, 260, 292, 288, 308, 300, 316, 312, 300, 332, 320, 364

Offset: 0

T. D. Noe, Nov 20 2002

nonn

Note that 2*A077768(n)-a(n)/4 is the characteristic sequence for the Beatty sequence A001951(n).

Eric Weisstein's World of Mathematics, Gauss's Circle Problem

Cf. A000328, A001951, A051132, A077768.

Table[Sum[SquaresR[2, k], {k, n^2 + 1, (n + 1)^2 - 1}], {n, 0, 100}]

a(n) = A051132(n+1) - A000328(n)

A093832 Values of r such that N(r)/r^2 > Pi, where N(r) is the number of integer lattice points (x,y) inside or on a circle of radius r.

Original entry on oeis.org

1, 2, 3, 5, 10, 15, 20, 35, 51, 52, 85, 100, 230, 247, 370, 425, 489, 725, 730, 1073, 1310, 1865, 1997, 2480, 2831, 3072, 3424, 3750, 3861, 3921, 4025, 4339, 4771, 4885, 5559, 5949, 6203, 6411, 7045, 7084, 7410, 7605, 8931, 9308, 9435, 9646, 10829, 10930

Offset: 1

Eric W. Weisstein, Apr 17 2004

nonn

David Wasserman, Table of n, a(n) for n = 1..160
Eric Weisstein's World of Mathematics, Gauss's Circle Problem.

Cf. A000328, A000328, A093837.

A000328(n) = local(x, y, c, nn); c = 0; x = 0; nn = n*n; y = n; while (x < y, c += x; y--; x = sqrtint(nn - y*y)); 4*(n - y) + 8*c + (2*y + 1)^2; for (n = 1, 100000, if (A000328(n) > Pi*n*n, print(n))); \\ David Wasserman, Dec 05 2006

Corrected and extended by David Wasserman, Dec 05 2006

Name corrected by Luis Mendo, Sep 24 2023

A093837 Denominator of N(n)/n^2, where N(n) is the number of lattice points (x,y) with x^2 + y^2 <= n^2.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 16, 169, 196, 225, 256, 17, 324, 361, 400, 441, 484, 529, 576, 625, 676, 243, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 432, 1369, 1444, 1521, 64, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 343, 500

Offset: 1

Eric W. Weisstein, Apr 17 2004

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gauss's Circle Problem

Cf. A093836 (numerators), A093832, A000328.

Definition edited (based on A093836) by Eric M. Schmidt, May 28 2015

A255238 Triangle T(n, m) of numbers of points of a square lattice covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m in the first quadrant.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 4, 3, 1, 6, 5, 5, 5, 4, 1, 7, 6, 6, 6, 5, 4, 1, 8, 7, 7, 7, 6, 5, 4, 1, 9, 8, 8, 8, 7, 7, 6, 4, 1, 10, 9, 9, 9, 9, 8, 7, 6, 5, 1, 11, 10, 10, 10, 10, 9, 9, 8, 7, 5, 1

Offset: 0

Wolfdieter Lang, Mar 12 2015

This entry is motivated by the proposal A255195 by Mats Granvik.
See the MathWorld link on Gauss's circle problem.
The first quadrant of a square lattice (x, y) with x = n >= 0, y = m >= 0, is considered. The number of lattice points 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, ..., n.
The same numbers occur if x and y are interchanged.
One could also consider the row reversed triangle.
The row sums give R(n) = A000603(n), n >= 0.
The alternating row sums give A255239(n), n >= 0.
The total number of square lattice points covered by a circular disk of radius n is A000328(n) = 4*R(n) - (4*n+3).

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

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

Cf. A000603, A000328, A255239.

T(n, m) = 1 + floor(sqrt(n^2 - m^2)), 0 <= m <= n.

cp's OEIS Frontend

A053411 Circle numbers (version 1): a(n)= number of points (i,j), i,j integers, contained in a circle of diameter n, centered at the origin.

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A049735 Array T(i,j) is the number of lattice points (x,y) in circle with radius (0,0)-to-(i,j), read by antidiagonals.

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A247588 Number of integer-sided acute triangles with largest side n.

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A036702 a(n)=number of Gaussian integers z=a+bi satisfying |z|<=n, a>=0, 0<=b<=a.

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A062711 Number of prime Gaussian integers z=a+bi with |z|<=n.

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A077770 Number of ordered pairs of integers (x,y) with n^2 < x^2 + y^2 < (n+1)^2; number of lattice points between circles of radii n and n+1.

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A093832 Values of r such that N(r)/r^2 > Pi, where N(r) is the number of integer lattice points (x,y) inside or on a circle of radius r.

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A093837 Denominator of N(n)/n^2, where N(n) is the number of lattice points (x,y) with x^2 + y^2 <= n^2.

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