A372219 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.

1, 12, 12, 17, 7, 30, 30, 43, 17, 56, 56, 81, 31, 90, 90, 131, 49, 132, 132, 193, 71, 182, 182, 267, 97, 240, 240, 353, 127, 306, 306, 451, 161, 380, 380, 561, 199, 462, 462, 683, 241, 552, 552, 817, 287, 650, 650, 963, 337, 756, 756, 1121, 391, 870, 870, 1291, 449, 992, 992, 1473

Offset

2,2

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.

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=2..61.

Miguel-Ángel Pérez García-Ortega, Teorema 10.12

Formula

Row n = (a, b, c, d) = (2n^2 - 1, 4n^2 + 6n + 2, 4n^2 + 6n + 2, 6n^2 + 8n + 3).

Example

Table begins:
  n=1:    1,  12,    12,    17;
  n=2:    7,  30,    30,    43;
  n=3:   17,  56,    56,    81;
  n=4:   31,  90,    90,   131;
  n=5:   49, 132,   132,   193;

Mathematica

cuaternas={}; Do[cuaternas=Join[cuaternas, {2n^2-1, 4n^2+6n+2, 4n^2+6n+2, 6n^2+8n+3}], {n, 1, 35}]; cuaternas

Crossrefs

Cf. A372220, A056220 (first column), A002939 (second column), A126587 (fourth column).

Sequence in context: A376819 A334620 A061074 * A064161 A260528 A260526

Adjacent sequences: A372216 A372217 A372218 * A372220 A372221 A372222

Keyword

nonn,easy,tabf

Author

Miguel-Ángel Pérez García-Ortega, Apr 22 2024

Status

approved