A382843 Length of the long leg in the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 = A000045(n) and its long leg and hypotenuse are consecutive natural numbers.

0, 0, 0, 4, 12, 40, 112, 312, 840, 2244, 5940, 15664, 41184, 108112, 283504, 742980, 1946364, 5097624, 13348944, 34953160, 91516920, 239607940, 627323620, 1642389984, 4299890112, 11257351200, 29472278112, 77159668612, 202007027820, 528861900424, 1384579459120

Offset

0,4

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.

Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Formula

a(n) = 2*Fibonacci(n)*(Fibonacci(n) - 1).

G.f.: 4*x^3/((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)). - Andrew Howroyd, Nov 12 2025

Example

Triples begin:
  n=0:     -1,     0,     1;
  n=1:      1,     0,     1;
  n=2:      1,     0,     1;
  n=3:      3,     4,     5.
This sequence gives column 2.

Mathematica

A382843[n_] := 2*#*(# - 1) & [Fibonacci[n]]; Array[A382843, 35, 0] (* or *)
LinearRecurrence[{3, 1, -5, -1, 1}, {0, 0, 0, 4, 12}, 35] (* Paolo Xausa, Jan 08 2026 *)

Prog

(PARI) a(n) = 2*fibonacci(n)*(fibonacci(n) - 1) \\ Andrew Howroyd, Nov 12 2025

Crossrefs

Cf. A000045, A382844 (area), A382845 (sum of the legs), A095122 (semiperimeter), A001595 (short leg).

Sequence in context: A248325 A344122 A328533 * A265127 A056274 A335806

Adjacent sequences: A382840 A382841 A382842 * A382844 A382845 A382846

Keyword

sign,easy

Author

Miguel-Ángel Pérez García-Ortega, Apr 06 2025

Status

approved