A371556 Table read by rows: row n is the unique primitive Pythagorean quadruple (a,b,c,d) such that (a + b + c - d)/2 = 2^n - 1 and a = b = 2^n.

4, 4, 7, 9, 8, 8, 31, 33, 16, 16, 127, 129, 32, 32, 511, 513, 64, 64, 2047, 2049, 128, 128, 8191, 8193, 256, 256, 32767, 32769, 512, 512, 131071, 131073, 1024, 1024, 524287, 524289, 2048, 2048, 2097151, 2097153, 4096, 4096, 8388607, 8388609, 8192, 8192, 33554431, 33554433, 16384, 16384, 134217727, 134217729

Offset

2,1

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..53.

Miguel-Ángel Pérez García-Ortega, Contando y calculando cuaternas pitagórcias.

Index entries for linear recurrences with constant coefficients, signature (-1, -1, -1, 6, 6, 6, 6, -8, -8, -8, -8).

Formula

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

Example

Table begins:
  n=2:    4,   4,     7,     9;
  n=3:    8,   8,    31,    33;
  n=4:   16,  16,   127,   129;
  n=5:   32,  32,   511,   513;
  n=6:   64,  64,  2047,  2049;

Mathematica

cuaternas={}; Do[cuaternas=Join[cuaternas, {2^n, 2^n, 2^(2n-1)-1, 2^(2n-1)+1}], {n, 2, 35}]; cuaternas

Crossrefs

Cf. A371559.

Sequence in context: A353713 A107432 A198991 * A214990 A185670 A011981

Adjacent sequences: A371553 A371554 A371555 * A371557 A371558 A371559

Keyword

nonn,easy,tabf

Author

Miguel-Ángel Pérez García-Ortega, Mar 27 2024

Status

approved