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

Author

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

Keywords

• easy
• nonn
• tabf

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.

Examples

			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;
  ...
		

References

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

Links

• 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.

Crossrefs

Cf. A001652, A001653, A046090.

Programs

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

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 *)

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) = (A002315(n) - 1)/2. - Miguel-Ángel Pérez García-Ortega, Nov 06 2024

Extensions

More terms from Robert Hutchins, Jun 10 2009

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