A024155 -id:A024155 - cp's OEIS frontend

A070141 Number of obtuse integer triangles with perimeter n having integral area.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 2

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051516(n) - A070140(n) - A024155(n).

Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Obtuse Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A051516, A070101, A070147.

A070205 Number of acute integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Acute Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A024155, A070093, A070140, A070201, A070206.

a(n) = A070201(n) - A024155(n) - A070206(n).

A070206 Number of obtuse integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070201(n) - A024155(n) - A070205(n).

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Obtuse Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070141, A070101.

A070207 Expansion of (1-x-5x^2)/(1-3x-2*x^2-x^3).

Original entry on oeis.org

1, 2, 3, 14, 50, 181, 657, 2383, 8644, 31355, 113736, 412562, 1496513, 5428399, 19690785, 71425666, 259086967, 939803018, 3409008654, 12365718965, 44854977221, 162705378247, 590191808148, 2140841158159, 7765612469020, 28168711531526, 102178200690777

Offset: 0

N. J. A. Sloane, Sep 18 2008

The old entry with this sequence number was a duplicate of A024155.

Benoit Rittaud, Elise Janvresse, Emmanuel Lesigne and Jean-Christophe Novelli, Quand les maths se font discrètes, Le Pommier, 2008 (ISBN 978-2-7465-0370-0). See pp. 42ff.

Robert Israel, Table of n, a(n) for n = 0..1770
Index entries for linear recurrences with constant coefficients, signature (3,2,1).

I:=[1,2,3]; [n le 3 select I[n] else 3*Self(n-1)+2*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 28 2015

f:= gfun:-rectoproc({-a(n+3)+3*a(n+2)+2*a(n+1)+a(n), a(0) = 1, a(1) = 2, a(2) = 3},a(n),remember):
map(f, [$0..50]); # Robert Israel, Dec 28 2015

CoefficientList[Series[(1-x-5x^2)/(1-3x-2x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,1},{1,2,3},40] (* Harvey P. Dale, Feb 01 2013 *)

Vec((1-x-5*x^2)/(1-3*x-2*x^2-x^3) + O(x^100)) \\ Altug Alkan, Dec 27 2015

a(0)=1, a(1)=2, a(2)=3, a(n) = 3*a(n-1)+2*a(n-2)+a(n-3). - Harvey P. Dale, Feb 01 2013

A299706 Number of Pythagorean triples with perimeter <= 10^n.

Original entry on oeis.org

0, 17, 325, 4858, 64741, 808950, 9706567, 113236940, 1294080089, 14557915466

Offset: 1

Seiichi Manyama, Feb 26 2018

			n = 2
perimeter | Pythagorean triple
-------------------------------
   12     | [ 3,  4,  5]
   30     | [ 5, 12, 13]
   24     | [ 6,  8, 10]
   56     | [ 7, 24, 25]
   40     | [ 8, 15, 17]
   36     | [ 9, 12, 15]
   90     | [ 9, 40, 41]
   60     | [10, 24, 26]
   48     | [12, 16, 20]
   84     | [12, 35, 37]
   60     | [15, 20, 25]
   90     | [15, 36, 39]
   80     | [16, 30, 34]
   72     | [18, 24, 30]
   70     | [20, 21, 29]
   84     | [21, 28, 35]
   96     | [24, 32, 40]

Eric Weisstein's World of Mathematics, Pythagorean Triple
Wikipedia, Tree of primitive Pythagorean triples

Cf. A024155, A101929, A101930, A101931, A249750.

def f(a, b, c, n)
  return 0 if a + b + c > n
  s = n / (a + b + c)
  s += f( a - 2 * b + 2 * c,  2 * a - b + 2 * c,  2 * a - 2 * b + 3 * c, n)
  s += f( a + 2 * b + 2 * c,  2 * a + b + 2 * c,  2 * a + 2 * b + 3 * c, n)
  s += f(-a + 2 * b + 2 * c, -2 * a + b + 2 * c, -2 * a + 2 * b + 3 * c, n)
  return s
end
def A299706(n)
  (1..n).map{|i| f(3, 4, 5, 10 ** i)}
end
p A299706(8)

A309395 Number of integer-sided triangles with sides a,b,c, max(a,b) < c, a + c = n that are right triangles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 3, 4, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 6, 1, 4

Offset: 1

Seiichi Manyama, Jul 28 2019

nonn

			   n | (a,b,c)
-----+-----------------------------------------
   8 | [ 3,  4,  5]
   9 | [ 4,  3,  5]
  16 | [ 6,  8, 10]
  18 | [ 5, 12, 13], [ 8,  6, 10]
  24 | [ 9, 12, 15]
  25 | [ 8, 15, 17], [12,  5, 13]
  27 | [12,  9, 15]
  32 | [ 7, 24, 25], [12, 16, 20], [15, 8, 17]
  36 | [10, 24, 26], [16, 12, 20]

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A004526, A024155, A046790, A082523.

{a(n) = sum(k=n\2+1, n-1, issquare(k^2-(n-k)^2))}

a(n) > 0 if and only if n is a term in A046790.

a(n^2) = floor((n-1)/2) = A004526(n-1).

cp's OEIS Frontend

A070141 Number of obtuse integer triangles with perimeter n having integral area.

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A070205 Number of acute integer triangles with perimeter n having integral inradius.

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A070206 Number of obtuse integer triangles with perimeter n having integral inradius.

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A070207 Expansion of (1-x-5*x^2)/(1-3*x-2*x^2-x^3).

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A299706 Number of Pythagorean triples with perimeter <= 10^n.

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Ruby

A309395 Number of integer-sided triangles with sides a,b,c, max(a,b) < c, a + c = n that are right triangles.

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Author

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Examples

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Programs

PARI

Formula

A070207 Expansion of (1-x-5x^2)/(1-3x-2*x^2-x^3).