A382379 Length of the long leg in the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

4, 0, 12, 24, 84, 220, 612, 1624, 4324, 11400, 30012, 78804, 206724, 541840, 1419612, 3718264, 9737284, 25496940, 66759012, 174788904, 457622004, 1198100200, 3136716012, 8212108324, 21499706884, 56287170720, 147362061612, 385799428824, 1010036895924

Offset

0,1

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*A000032(n)*(A000032(n) - 1).

G.f.: 4*(1 - 3*x + 2*x^2 + 2*x^3 + x^4)/((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)). - Andrew Howroyd, Nov 16 2025

Example

The triangles begin:
  n=0:      3,     4,     5;
  n=1:      1,     0,     1;
  n=2:      5,    12,    13;
  n=3:      7,    24,    25;
  ...
This sequence gives the middle column

Mathematica

A382379[n_] := 2*#*(# - 1) & [LucasL[n]]; Array[A382379, 30, 0] (* or *)
LinearRecurrence[{3, 1, -5, -1, 1}, {4, 0, 12, 24, 84}, 30] (* Paolo Xausa, Jan 08 2026 *)

Prog

(PARI) a(n) = my(t=fibonacci(n+1)+fibonacci(n-1)); 2*t*(t-1); \\ Andrew Howroyd, Nov 16 2025

Crossrefs

Cf. A000032, A022319 (short leg), A382409 (semiperimeter), A382410 (area), A386201.

Sequence in context: A273682 A170878 A056460 * A137523 A375988 A369914

Adjacent sequences: A382376 A382377 A382378 * A382380 A382381 A382382

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Mar 24 2025

Status

approved