A378395 Sequence of primitive Pythagorean triples beginning with the triple (3,4,5), with each subsequent triple having as its inradius the hypotenuse of the previous triple, and with the long leg and the hypotenuse of each triple being consecutive natural numbers.

3, 4, 5, 11, 60, 61, 123, 7564, 7565, 15131, 114473580, 114473581, 228947163, 26208401722874284, 26208401722874285, 52416803445748571, 1373760641735119632984407274271020, 1373760641735119632984407274271021

Offset

1,1

Comments

The only Pythagorean triple whose inradius is equal to r and such that its long leg and its hypotenuse are consecutive is (2r+1,2r^2+2r,2r^2+2r+1).

References

J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2024.

Links

Table of n, a(n) for n=1..18.

Example

Triples begin:
  3, 4, 5;
  11, 60, 61;
  123, 7564, 7565;
  15131, 114473580, 11447358;
...

Mathematica

{a0, b0, c0}={3, 4, 5}; f[n_]:=Module[{fn0=2c0+1, fn1=((2c0+1)^2+1)/2}, Do[{fn0, fn1}={2fn1+1, ((2fn1+1)^2+1)/2}, {n}]; fn0]; t[n_]:={f[n-1], (f[n-1]^2-1)/2, (f[n-1]^2+1)/2}; ternas={a0, b0, c0}; For[i=1, i<=6, i++, ternas=Join[ternas, t[i]]]; ternas

Crossrefs

Cf. A102847 (short leg), A365577, A365578, A365796.

Sequence in context: A341785 A052276 A173096 * A302752 A046964 A296966

Adjacent sequences: A378392 A378393 A378394 * A378396 A378397 A378398

Keyword

nonn,tabf

Author

Miguel-Ángel Pérez García-Ortega, Nov 24 2024

Status

approved