A371559 Table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that (a + b + c - d)/2 = n and b - a = d - c = 1.

1, 2, 2, 3, 2, 3, 6, 7, 3, 4, 12, 13, 4, 5, 20, 21, 5, 6, 30, 31, 6, 7, 42, 43, 7, 8, 56, 57, 8, 9, 72, 73, 9, 10, 90, 91, 10, 11, 110, 111, 11, 12, 132, 133, 12, 13, 156, 157, 13, 14, 182, 183, 14, 15, 210, 211, 15, 16, 240, 241, 16, 17, 272, 273, 17, 18, 306, 307, 18, 19, 342, 343, 19, 20, 380, 381, 20, 21, 420, 421

Offset

1,2

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=1..80.

Miguel-Ángel Pérez García-Ortega, Contando y calculando cuaternas pitagóricas

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,2,-2,2,-2,-1,1,-1,1).

Formula

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

G.f.: -x*(x^10-x^9+2*x^8-2*x^7-3*x^4+2*x^3+x^2+x+1)/((x+1)^2*(x-1)^3*(x^2+1)^3). - Alois P. Heinz, Sep 18 2024

Example

Table begins:
  n=1:   1,   2,   2,   3;
  n=2:   2,   3,   6,   7;
  n=3:   3,   4,  12,  13;
  n=4:   4,   5,  20,  21;
  n=5:   6,   7,  42,  43;

Mathematica

cuaternas={}; Do[cuaternas=Join[cuaternas, {n, n+1, n^2+n, n^2+n+1}], {n, 1, 35}]; cuaternas
Table[{n, n+1, n^2+n, n^2+n+1}, {n, 20}]//Flatten (* Harvey P. Dale, Sep 06 2024 *)

Crossrefs

Cf. A371556.

Sequence in context: A129568 A177892 A341758 * A236433 A117122 A122828

Adjacent sequences: A371556 A371557 A371558 * A371560 A371561 A371562

Keyword

nonn,easy,tabf

Author

Miguel-Ángel Pérez García-Ortega, Mar 27 2024

Status

approved