A249750 -id:A249750 - cp's OEIS frontend

A118858 Decimal expansion of log(2)/Pi^2.

0, 7, 0, 2, 3, 0, 4, 9, 2, 7, 7, 2, 6, 8, 2, 8, 7, 6, 4, 0, 8, 9, 3, 8, 5, 9, 9, 4, 9, 6, 9, 9, 7, 0, 0, 9, 6, 3, 2, 8, 7, 6, 5, 3, 2, 4, 4, 3, 2, 6, 2, 5, 4, 1, 3, 7, 7, 4, 3, 4, 3, 7, 8, 2, 2, 8, 2, 4, 9, 6, 4, 1, 3, 3, 6, 9, 6, 8, 5, 3, 4, 0, 1, 4, 2, 0, 1, 6, 9, 3, 5, 8, 0, 7, 3, 3, 4, 0, 0, 9, 4, 3, 2, 7, 6

Offset: 0

Eric W. Weisstein, May 02 2006

			0.07023049277268287640...

Derrick Norman Lehmer, Asymptotic Evaluation of Certain Totient Sums, American Journal of Mathematics, Vol. 22, No. 4 (1900), pp. 293-335.
Eric Weisstein's World of Mathematics, Primitive Pythagorean Triple.

Cf. A249750, A328499, A002162, A002388, A284983.

RealDigits[Log[2]/Pi^2, 10, 100][[1]] (* Amiram Eldar, Jun 25 2021 *)

log(2)/Pi^2 \\ Michel Marcus, Jun 25 2021

Equals lim_{n->oo} A328499(n)/n. - Amiram Eldar, Jun 25 2021

Equals Integral_{z>=0} z*sech(Pi*z)^2. - Peter Luschny, Aug 03 2021

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)

A328499 The number of primitive Pythagorean triangles with perimeter less than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Mo Li, Oct 17 2019

D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = log(2)/Pi^2 = 0.07023049... See A118858.

			For n=90, the triples are
   {3,  4,  5},  3 +  4 +  5 = 12 < 90
   {5, 12, 13},  5 + 12 + 13 = 30 < 90
   {7, 24, 25},  7 + 24 + 25 = 56 < 90
   {8, 15, 17},  8 + 15 + 17 = 40 < 90
   {9, 40, 41},  9 + 40 + 41 = 90
  {12, 35, 37}, 12 + 35 + 37 = 84 < 90
  {20, 21, 29}, 20 + 21 + 29 = 70 < 90
so a(90)=7.

Ron Knott, Pythagorean Triples and Online Calculators
D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.

Cf. A008846, A024364, A118858, A249750.

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A118858 Decimal expansion of log(2)/Pi^2.

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

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A328499 The number of primitive Pythagorean triangles with perimeter less than n.

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