A382608 Long leg of the unique primitive Pythagorean triple whose inradius is A000045(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

0, 4, 4, 12, 24, 60, 144, 364, 924, 2380, 6160, 16020, 41760, 109044, 285012, 745420, 1950312, 5104012, 13359280, 34969884, 91543980, 239651724, 627394464, 1642504612, 4300075584, 11257651300, 29472763684, 77160454284, 202008299064, 528863957340, 1384582787280

Offset

0,2

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*F(n)*(F(n) + 1) where F(n) = A000045(n).

From Andrew Howroyd, Nov 13 2025: (Start)

a(n) = 4*A033192(n).

G.f.: 4*x*(1 - 2*x - x^2 + x^3)/((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)). (End)

Example

Triangle begins:
  n=1:      3,    4,    5;
  n=2:      3,    4,    5;
  n=3:      5,   12,   13;
where this sequence is the middle column.

Mathematica

a=Table[Fibonacci[n], {n, 0, 16}]; Apply[Join, Map[{2#+1, 2#^2+2#, 2#^2+2#+1}&, a]]

Prog

(PARI) a(n) = 2*fibonacci(n)*(fibonacci(n) + 1); \\ Andrew Howroyd, Nov 13 2025

Crossrefs

Cf. A000045 (inradius), A001588 (short leg), A382609 (semiperimeter), A382610 (area), A033192.

Sequence in context: A303644 A298796 A106232 * A359709 A228612 A038804

Adjacent sequences: A382605 A382606 A382607 * A382609 A382610 A382611

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Mar 31 2025

Extensions

a(0)=0 prepended by Andrew Howroyd, Nov 13 2025

Status

approved