A247588 -id:A247588 - cp's OEIS frontend

A000328 Number of points of norm <= n^2 in square lattice.

1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441, 529, 613, 709, 797, 901, 1009, 1129, 1257, 1373, 1517, 1653, 1793, 1961, 2121, 2289, 2453, 2629, 2821, 3001, 3209, 3409, 3625, 3853, 4053, 4293, 4513, 4777, 5025, 5261, 5525, 5789, 6077, 6361, 6625

Offset: 0

N. J. A. Sloane

Number of ordered pairs of integers (x,y) with x^2 + y^2 <= n^2.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
H. Gupta, A Table of Values of N_3(t), Proc. National Institute of Sciences of India, 13 (1947), 35-63.
C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 47.
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 and Robert Israel, Table of n, a(n) for n = 0..10000 (n=0..1000 from T. D. Noe)
W. Fraser and C. C. Gotlieb, A calculation of the number of lattice points in the circle and sphere, Math. Comp., 16 (1962), 282-290.
Eric Weisstein's World of Mathematics, Gauss's Circle Problem
Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025. See Table 3 p. 18.

Column k=2 of A302997.
Equals A051132 + A046109. For another version see A057655.
Cf. A093832, A093836, A093837, A000603, A255238, A305575, A305576.

a000328 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 <= n^2]
-- Reinhard Zumkeller, Jan 23 2012

Table[Sum[SquaresR[2, k], {k, 0, n^2}], {n, 0, 46}]

{ a(n) = 1 + 4 * sum(j=0,n^2\4, n^2\(4*j+1) - n^2\(4*j+3) ) } /* Max Alekseyev, Nov 18 2007 */

def A000328(n):
    return (sum([int((n**2 - y**2)**0.5) for y in range(1, n)]) * 4 + 4*n + 1)
    # Karl-Heinz Hofmann, Aug 03 2022

from math import isqrt
def A000328(n): return 1+(sum(isqrt(k*((n<<1)-k)) for k in range(1,n+1))<<2) # Chai Wah Wu, Feb 12 2025

a(n) = 1 + 4 * Sum_{j>=0} floor(n^2/(4*j+1)) - floor(n^2/(4*j+3)). Also a(n) = A057655(n^2). - Max Alekseyev, Nov 18 2007
a(n) = 4*A000603(n) - (4*n+3), n >= 0. - Wolfdieter Lang, Mar 15 2015
a(n) = 1+4*n^2-4*ceiling((n-1)/sqrt(2))-8*A247588(n-1), n>1. - Mats Granvik, May 23 2015
a(n) = [x^(n^2)] theta_3(x)^2/(1 - x), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 14 2018
Limit_{n->oo} a(n)/n^2 = Pi. - Chai Wah Wu, Feb 12 2025

More terms from David W. Wilson, May 22 2000
Edited at the suggestion of Max Alekseyev by N. J. A. Sloane, Nov 18 2007
Incorrect comment removed by Eric M. Schmidt, May 28 2015

A247587 Number of obtuse triangles with integer sides at most n.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 13, 20, 30, 42, 57, 74, 95, 120, 149, 182, 219, 261, 309, 362, 420, 485, 556, 632, 715, 806, 906, 1012, 1125, 1247, 1377, 1517, 1666, 1824, 1993, 2170, 2358, 2555, 2765, 2986

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(4) = 2 because there are 2 obtuse triangles with integer sides less than or equal to 4: (2,2,3); (2,3,4).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vladimir Letsko, Mathematical Marathon, problem 192 (in Russian).

Cf. A002623, A224921, A247586, A247588, A247589.

tr_o:=proc(n) local a, b, c, t, d; t:=0:
  for a to n do
  for b from a to n do
  for c from b to min(a+b-1, n) do
  d:=a^2+b^2-c^2:
  if d<0 then t:=t+1 fi
  od od od;
  [n, t]; end;

a(n)=sum(a=2,n-1,sum(b=a,n-1,max(0,min(n,a+b-1)-sqrtint(a^2+b^2)))) \\ Charles R Greathouse IV, Sep 20 2014

obtuse(n)=sum(a=2,n-1, max(0, sqrtint(n^2-1-a^2)-max(a,n-a+1)+1))
s=0; vector(100,n, s+=obtuse(n)) \\ Charles R Greathouse IV, Sep 20 2014

A236384 Number of non-congruent integer triangles with base length n whose apex lies on or within a space bounded by a semicircle of diameter n.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 5, 7, 10, 13, 15, 17, 22, 25, 30, 33, 38, 42, 48, 54, 58, 65, 71, 76, 85, 92, 100, 106, 114, 123, 130, 140, 149, 159, 170, 177, 189, 197, 211, 222, 231, 243, 255, 269, 282, 292, 306, 318, 333, 348, 364, 378, 391, 406, 420, 438, 453, 470, 485

Offset: 1

Frank M Jackson, Jan 24 2014

nonn

Number of integer-sided obtuse or right (non-acute) triangles with largest side n. - Frank M Jackson, Dec 03 2014

			a(5)=3 as there are 3 non-congruent integer triangles with base length 5 whose apex lies on or within the space bounded by the semicircle of diameter 5. The integer triples are (2,4,5), (3,3,5), (3,4,5).

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A002620, A247588.

c = Slider(1, 20, 1);
L = Flatten(Sequence(Sequence(((a^2+c^2-(c-a+k)^2)/(2c),((a+(c-a+k)+c)(a+(c-a+k)-c)(a-(c-a+k)+c)(-a+(c-a+k)+c))^(1/2)/(2c)),a,k,(c+k)/2),k,1,c));
C = {Circle((c/2,0),c/2)};
a_n = CountIf(IsInRegion((x(A),y(A)),Element(C,1)),A,L);
# Frank M Jackson, Jan 01 2024

sumtriangles[c_] := (n=0; Do[If[a^2+b^2<=c^2, n++], {b, 1, c}, {a, c-b+1, b}]; n); Table[sumtriangles[m], {m, 1, 200}]

a(n)=sum(a=2,n,sum(b=max(a,n+1-a),n,a^2+b^2<=n^2)) \\ Charles R Greathouse IV, Mar 26 2014

a(n)=sum(a=2,n,max(min(sqrtint(n^2-a^2),n)-max(a,n+1-a)+1,0)) \\ Charles R Greathouse IV, Mar 26 2014

a(n) = A002620(n+1)-A247588(n). - Frank M Jackson, Dec 03 2014

A247586 Number of acute triangles with integer sides less than or equal to n.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 36, 49, 64, 81, 102, 127, 154, 185, 219, 258, 301, 349, 401, 457, 520, 587, 660, 740, 824, 914, 1010, 1114, 1225, 1342, 1468, 1600, 1740, 1887, 2041, 2206, 2378, 2561, 2750, 2948

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(2) = 3 because there are 3 acute triangles with integer sides less than or equal to 2: (1,1,1); (1,2,2); (2,2,2).

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

Cf. A002623, A224921, A247587, A247588, A247589.

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

a[n_] := Module[{a, b, c, d, t = 0}, Do[d = a^2 + b^2 - c^2; If[d>0, t++], {a, n}, {b, a, n}, {c, b, Min[a+b-1, n]}]; t]; Array[a, 40] (* Jean-François Alcover, Jun 19 2019, from Maple *)

import itertools
def A247586(n):
    I = itertools.combinations_with_replacement(range(1,n+1),3)
    F = filter(lambda c: c[0]**2 + c[1]**2 > c[2]**2, I)
    return len(list(F))
print([A247586(n) for n in range(41)]) # Peter Luschny, Sep 22 2014

A255195 Triangle describing the shape of one eighth of the Gauss circle problem.

Original entry on oeis.org

1, 2, 0, 2, 1, 0, 2, 1, 1, 0, 2, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 1, 2, 2, 0, 0, 0, 2, 1, 1, 2, 1, 2, 0, 0, 0, 2, 1, 1, 1, 2, 2, 1, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0

Offset: 1

Mats Granvik, Feb 16 2015

The sum of terms of row n is n.

Total of partial sums in reverse (from right to left) equals one eighth of the Gauss circle problem. Whenever there is the number 2 the border of the circle makes a jump upwards. Predicting where the 2's are would say something about the Gauss circle problem. The number of 2's equals the number of 0's in the same row, and is counted by A194920(n-1).

			1,
2, 0,
2, 1, 0,
2, 1, 1, 0,
2, 1, 2, 0, 0,
2, 1, 1, 2, 0, 0,
2, 1, 1, 2, 1, 0, 0,
2, 1, 1, 2, 2, 0, 0, 0,
2, 1, 1, 2, 1, 2, 0, 0, 0,
2, 1, 1, 1, 2, 2, 1, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0

E. W. Weisstein, Gauss's Circle Problem

Cf. A000603, A194920.

Flatten[Table[Sum[Table[If[And[If[n^2 + k^2 <= r^2, If[n >= k, 1, 0], 0] == 1, If[(n + 1)^2 + (k + 1)^2 <= r^2, If[n >= k, 1, 0], 0]== 0], 1, 0], {k, 0, r}], {n, 0, r}], {r, 0, 12}]]

A000603(n) = 2*(Sum_{k=1..n} Sum_{k=1..k} T(n,n-k+1))-ceiling((n-1)/sqrt(2)) for n>1.

A247588(n-1) = (Sum_{k=1..n} Sum_{k=1..k} (T(n,k) - T(n,n-k+1))/2).

A306673 a(n) is the number of distinct, non-similar acute triangles with integer sides and largest side <= n.

Original entry on oeis.org

1, 2, 4, 7, 12, 16, 26, 34, 46, 56, 76, 90, 116, 135, 161, 187, 229, 257, 308, 344, 394, 439, 511, 558, 636, 698, 779, 849, 959, 1027, 1152, 1245, 1362, 1465, 1603, 1703, 1874, 2004, 2164, 2298, 2507, 2639, 2867, 3034, 3235, 3421, 3690, 3866, 4147, 4354

Offset: 1

César Eliud Lozada, Mar 04 2019

nonn

			For n=4, there are 9 acute triangles with integer sides and largest side <= 4. These have sides {a,b,c} = {1, 1, 1}, {1, 2, 2}, {1, 3, 3}, {1, 4, 4}, {2, 2, 2}, {2, 2, 4}, {2, 3, 3}, {3, 3, 4}, {3, 4, 4}. But {2, 2, 2} is similar to {1,1,1} and {2,2,4} is similar to {1,1,2}, so these two triangles are excluded from the list and therefore a(4)=7.

Cf. A247588, A306674, A306676, A306677, A306678.

#nType=1 for acute triangles, nType=2 for obtuse triangles, nType=0 for both triangles
CountTriangles := proc (n, nType := 1)
  local aa, oo, a, b, c, tt, lAcute;
  aa := {}; oo := {};
  for a from n by -1 to 1 do for b from a by -1 to 1 do for c from b by -1 to 1 do
    if a < b+c and abs(b-c) < a and b < c+a and abs(c-a) < b and c < a+b and abs(a-b) < c and gcd(a, gcd(b, c)) = 1 then
      lAcute := evalb(0 < b^2+c^2-a^2);
      tt := sort([a, b, c]);
      if lAcute then aa := {op(aa), tt} else oo := {op(oo), tt} end if
    end if
  end do end do end do;
  return sort(`if`(nType = 1, aa, `if`(nType=2,oo,`union`(aa,oo))))
end proc

Length@Select[DeleteDuplicates[Sort@# & /@ Tuples[Range@#, 3]], GCD @@ # == 1 && #[[1]] + #[[2]] > #[[3]] && #[[1]]^2 + #[[2]]^2 > #[[3]]^2 &] & /@ Range@50 (* Hans Rudolf Widmer, Dec 07 2023 *)

A247589 Number of integer-sided obtuse triangles with largest side n.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 7, 10, 12, 15, 17, 21, 25, 29, 33, 37, 42, 48, 53, 58, 65, 71, 76, 83, 91, 100, 106, 113, 122, 130, 140, 149, 158, 169, 177, 188, 197, 210, 221, 230, 243, 255, 269, 281, 292, 306, 318, 333, 346

Offset: 1

Vladimir Letsko, Sep 20 2014

nonn

			a(5) = 2 because there are 2 integer-sided acute triangles with largest side 5: (2,4,5); (3,3,5).

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

Cf. A046080, A002623, A224921, A247586, A247587, A247588.

tr_o:=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) = k*(k + (1+(-1)^n)/2) + Sum_{j=1..floor(n*(1-sqrt(2)/2))} floor(sqrt(2*j*n - j^2 - 1) - j), where k = floor((2*n*(sqrt(2) - 1) + 1 - (-1)^n)/4) (it appears that k(n) is A070098(n)). - Anton Nikonov, Sep 29 2014

A251607 Number of non-congruent integer triangles ABC with largest side BC of length n whose opposite vertex A lies inside the closed boundary formed by BC and a locus of points whose real distance from B is x, from C is y and x^3 + y^3 = n^3.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 10, 14, 16, 20, 24, 29, 35, 39, 45, 51, 58, 66, 71, 80, 87, 97, 104, 115, 124, 133, 145, 154, 166, 178, 189, 202, 213, 228, 241, 254, 269, 283, 297, 315, 329, 346, 361, 380, 396, 413, 431, 450, 470, 487, 508, 528, 548, 569, 588, 613, 632, 655, 677, 701

Offset: 1

Frank M Jackson, Dec 05 2014

nonn

Acute integer triangles ABC with longest side BC of length n (A247588) are segregated from obtuse or right integer triangles with the same longest side BC (A236384) by the closed boundary formed by a semicircle and BC as its diameter. The right integer triangles will lie on this boundary and the obtuse integer triangles within this boundary. Define a closed boundary S(q) that is formed by BC and a locus of points whose real distance from B is x, from C is y and x^q + y^q = n^q for integer q > 0. Then S(2) is that closed boundary formed by a semicircle with BC as diameter. Euler proved that there are no integer triangles that lie on S(3) and Wiles for all S(q) where q > 2. This sequence identifies all integer triangles with longest side BC of length n that lie inside S(3).

			a(5)=3 as there are 3 non-congruent integer triangles with base length of 5 whose apex lies inside S(3). The integer triples are (3,3,5), (2,4,5), (3,4,5). The other triangles from the complete set of non-congruent integer triangles with longest side length 5 (A002620(5+1)) are (4,4,5), (1,5,5), (2,5,5), (3,5,5), (4,5,5), (5,5,5) and lie outside the closed boundary.

Cf. A002620, A236384, A247588.

n = Slider(1, 20, 1);
L = Flatten(Sequence(Sequence(((a^2+n^2-(n-a+k)^2)/(2n),((a+(n-a+k)+n)(a+(n-a+k)-n)(a-(n-a+k)+n)(-a+(n-a+k)+n))^(1/2)/(2n)),a,k,(n+k)/2),k,1,n));
p = 3;
C = Curve((a^2+n^2-(n^p-a^p)^(2/p))/(2n),((a+(n^p-a^p)^(1/p)+n)(a+(n^p-a^p)^(1/p)-n)(a-(n^p-a^p)^(1/p)+n)(-a+(n^p-a^p)^(1/p)+n))^(1/2)/(2n),a,0,n);
a_n = CountIf((x(A)^2+y(A)^2)^(p/2)+((n-x(A))^2+y(A)^2)^(p/2)Frank M Jackson, Jan 02 2024

sumtriangles[c_] := (n = 0; Do[If[a^3+b^3c, n++], {b, 1, c}, {a, 1, b}]; n); Table[sumtriangles[m], {m, 1, 200}]

cp's OEIS Frontend

A000328 Number of points of norm <= n^2 in square lattice.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Python

Formula

Extensions

A247587 Number of obtuse triangles with integer sides at most n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

PARI

A236384 Number of non-congruent integer triangles with base length n whose apex lies on or within a space bounded by a semicircle of diameter n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GeoGebra

Mathematica

PARI

PARI

Formula

A247586 Number of acute triangles with integer sides less than or equal to n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A255195 Triangle describing the shape of one eighth of the Gauss circle problem.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A306673 a(n) is the number of distinct, non-similar acute triangles with integer sides and largest side <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A247589 Number of integer-sided obtuse triangles with largest side n.

Views

Author

Keywords

Examples

Links

Crossrefs