cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Vladimir Letsko, Sep 20 2014

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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;
  • Mathematica
    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 *)
  • PARI
    a(n) = sum(j=0, n*(1 - sqrt(2)/2), n - j - floor(sqrt(2*j*n - j^2))); \\ Michel Marcus, Oct 07 2014
  • PARI
    {a(n) = sum(j=0, n - sqrtint(n*n\2) - 1, n - j - sqrtint(2*j*n - j*j))}; /* Michael Somos, May 24 2015 */

Formula

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

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

Views

Author

Vladimir Letsko, Sep 20 2014

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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;
  • PARI
    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
  • PARI
    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

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

Views

Author

Vladimir Letsko, Sep 20 2014

Keywords

Examples

			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).

Links

Crossrefs

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

Programs

  • Maple
    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;
  • Mathematica
    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 *)
  • Python
    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
Showing 1-3 of 3 results.

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