A114336 Pythagorean triples of nearly isosceles triangle.

3, 4, 5, 20, 21, 29, 119, 120, 169, 696, 697, 985, 4059, 4060, 5741, 23660, 23661, 33461, 137903, 137904, 195025, 803760, 803761, 1136689, 4684659, 4684660, 6625109, 27304196, 27304197, 38613965, 159140519, 159140520, 225058681, 927538920, 927538921, 1311738121, 5406093003, 5406093004, 7645370045

Offset

1,1

Comments

Pythagorean triples of exact isosceles triangles do not exist because 2a^2 = c^2 has no integer solution. a^2 + (a+1)^2 = c^2 are nearly isosceles triangles and give a recursive series.

References

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Links

Table of n, a(n) for n=1..39.

C.C. Chen and T.A. Peng, Classroom note: Almost-isosceles right-angled triangles, Australasian Journal of Combinatorics, Volume 11(1995), pp. 263-267. See p. 266.

Formula

a^2 + (a+1)^2 = c^2, a(n) = 3a(n-1) + 2c(n-1) + 1, c(n) = 4a(n-1) + 3c(n-1) + 2.

a(n) = (A377011(n,1) - 1)/2 and c(n) = sqrt(A377011(n,3)). - Miguel-Ángel Pérez García-Ortega, Nov 06 2024

Example

119^2 + 120^2 = 169^2.
Triples begin:
  n=1:   3,   4,   5;
  n=2:  20,  21,  29;
  n=3: 119, 120, 169;
  n=4: 696, 697, 985;
  ...

Mathematica

a=Table[(LucasL[2*n+1, 2]-2)/4, {n, 1, 13}]; Apply[Join, Map[{#, #+1, Sqrt[2#^2+2#+1]}&, a]] (* Miguel-Ángel Pérez García-Ortega, Nov 06 2024 *)

Prog

(BASIC)
a(1):= 3
c(1):= 5
for n:=2 until 10 step 1
a(n):= 3*a(n-1) + 2*c(n-1) + 1
c(n):= 4*a(n-1) + 3*c(n-1) + 2
print a(n), a(n)+1, c(n)
next n
end

Crossrefs

Cf. A001652, A001653, A046090, A377011.

Sequence in context: A161474 A247573 A084930 * A048086 A048005 A334638

Adjacent sequences: A114333 A114334 A114335 * A114337 A114338 A114339

Keyword

easy,nonn,tabf

Author

Heinrich Baldauf (heinbald25(AT)web.de), Feb 07 2006

Extensions

More terms from Robert Hutchins, Jun 10 2009

More terms from Miguel-Ángel Pérez García-Ortega, Nov 06 2024

Status

approved