A101931 -id:A101931 - cp's OEIS frontend

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

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)

A379744 Number of primitive Pythagorean quintuples (a, b, c, d, e) with 0 < a <= b <= c <= d <= e <= 10^n.

Original entry on oeis.org

10, 5568, 5302303, 5279762116, 5277410421368, 5277177914347752, 5277147974562930196, 5277145259376056385184, 5277145005746992952994327

Offset: 1

Asif Ahmed, Dec 31 2024

A Pythagorean quintuple (x,y,z,w,u) is a solution to x^2+y^2+z^2+w^2=u^2.

			a(1) = 10 because there are ten primitive solutions (a, b, c, d, e) as follows: (1, 1, 1, 1, 2), (1, 1, 3, 5, 6), (1, 1, 7, 7, 10), (1, 2, 2, 4, 5), (1, 3, 3, 9, 10), (1, 4, 4, 4, 7), (1, 5, 5, 7, 10), (2, 2, 3, 8, 9), (2, 2, 4, 5, 7), and (2, 4, 5, 6, 9) with e <= 10.

Math Stack Exchange, Pythagorean quintuple x^2+y^2+z^2+w^2=u^2 has infinite solutions. Is it difficult to find the general formula of its integer solution?.

Cf. A101931, A157085, A225207.

Limit_{n -> oo} a(n)/ 10^(3*n) = 5/(96*Pi^2) ~ 0.005277144981371758929368722042173314526269...

a(n) ~ 5*10^(3*n)/(96*Pi^2) + (3/A - 1/G)*10^(2*n)/64 + (1/(2*sqrt(3)) - 1/(4*sqrt(2)))*10^n/Pi, where A is the Dirichlet L-function value evaluated at s = 2 for the Dirichlet character with modulus 8 and index 4, and G is the Catalan's constant. (A ~ 1.064734171043503370392827451461668889483, G ~ 0.9159655941772190150546035149323841107741)

cp's OEIS Frontend

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

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