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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.
2
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
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
KEYWORD
nonn,easy,tabf
STATUS
approved