A372220 Four-column table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that a > (a + b + c - d)/2 = 2n(n + 1) and b = c.

17, 20, 20, 33, 31, 42, 42, 67, 49, 72, 72, 113, 71, 110, 110, 171, 97, 156, 156, 241, 127, 210, 210, 323, 161, 272, 272, 417, 199, 342, 342, 523, 241, 420, 420, 641, 287, 506, 506, 771, 337, 600, 600, 913, 391, 702, 702, 1067, 449, 812, 812, 1233, 511, 930, 930, 1411, 577, 1056, 1056, 1601

Offset: 2

Author

Miguel-Ángel Pérez García-Ortega, Apr 22 2024

Keywords

• nonn
• easy
• tabf

Comments

A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.

Examples

			Table begins:
  n=2:   17,   20,    20,    33;
  n=3:   31,   42,    42,    67;
  n=4:   49,   72,    72,   113;
  n=5:   71,  110,   110,   171;
  n=6:   97,  156,   156,   241;
		

References

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

Links

• Miguel-Ángel Pérez García-Ortega, Teorema 10.13

Crossrefs

Cf. A372219, A056220 (first column), A002943 (second column), A080859 (fourth column).

Programs

Mathematica

cuaternas={};Do[cuaternas=Join[cuaternas,{2n^2+4n+1,4n^2+2n,4n^2+2n,6n^2+4n+1}],{n,2,35}];cuaternas

Formula

Row n = (a, b, c, d) = (2n^2 + 4n + 1, 4n^2 + 2n, 4n^2 + 2n, 6n^2 + 4n + 1).