A382845 Sum of the legs of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 = A000045(n) and its long leg and hypotenuse are consecutive natural numbers.

-1, 1, 1, 7, 17, 49, 127, 337, 881, 2311, 6049, 15841, 41471, 108577, 284257, 744199, 1948337, 5100817, 13354111, 34961521, 91530449, 239629831, 627359041, 1642447297, 4299982847, 11257501249, 29472520897, 77160061447, 202007663441, 528862928881, 1384581123199

Offset

0,4

References

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

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 (3,0,-3,1).

Formula

a(n) = A382843(n,1) + A382843(n,2).

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

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

Example

For n=4, the short leg is A382843(2,1) = 3 and the long leg is A382843(2,2) = 4  so the sum of the legs is then a(4) = 3 + 4 = 7.

Mathematica

a=Table[Fibonacci[n], {n, 0, 30}]; Apply[Join, Map[{2#^2-1}&, a]]

Prog

(PARI) a(n) = 2*fibonacci(n)^2 - 1

Crossrefs

Cf. A000045, A382843, A382844, A095122, A007598, A080097.

Sequence in context: A110097 A179262 A018672 * A221591 A045821 A262754

Adjacent sequences: A382842 A382843 A382844 * A382846 A382847 A382848

Keyword

sign,easy

Author

Miguel-Ángel Pérez García-Ortega, Apr 06 2025

Status

approved