A384288 Length of the long leg in the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

0, 12, 84, 312, 840, 1860, 3612, 6384, 10512, 16380, 24420, 35112, 48984, 66612, 88620, 115680, 148512, 187884, 234612, 289560, 353640, 427812, 513084, 610512, 721200, 846300, 987012, 1144584, 1320312, 1515540, 1731660, 1970112, 2232384, 2520012, 2834580

Offset

0,2

Links

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

José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 2 * A002378(n) * (A002378(n) + 1).

From Andrew Howroyd, Nov 12 2025: (Start)

a(n) = 12*A006325(n + 1).

a(n) = 2*n*(n + 1)*(n^2 + n + 1).

G.f.: 12*x*(1 + x)^2/(1 - x)^5. (End)

Example

Triangles begin:
  n=1:      5,   12,   13;
  n=2:     13,   84,   85;
  n=3:     25,  312,  313;
  ...
This sequence gives the middle column.

Mathematica

A384288[n_] := 2*n*(n + 1)*(n^2 + n + 1); Array[A384288, 50, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 12, 84, 312, 840}, 50] (* Paolo Xausa, Jan 07 2026 *)

Crossrefs

Cf. A002378 (inradius), A001844 (short leg), A008514 (sum of the legs), A237516 (semiperimeter), A384566 (area), A006325.

Sequence in context: A392988 A213784 A085409 * A303916 A111464 A341367

Adjacent sequences: A384285 A384286 A384287 * A384289 A384290 A384291

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, May 31 2025

Extensions

a(0)=0 prepended by Andrew Howroyd, Nov 12 2025

Status

approved