A377016 Semiperimeter of the unique primitive Pythagorean triple whose short leg is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

1, 28, 861, 28680, 970921, 32963140, 1119662181, 38034888528, 1292062686481, 43892073946540, 1491038320325421, 50651410052600280, 1720656899012149561, 58451683130389395028, 1985636569382856677301, 67453191675004485098400, 2291422880375627492063521, 77840924741066359629967420

Offset

0,2

Comments

a(0) = 1 is included by convention. This corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.

References

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Links

Table of n, a(n) for n=0..17.

Index entries for linear recurrences with constant coefficients, signature (41,-246,246,-41,1).

Formula

a(n) = (A377011(n,1) + A377011(n,2) + A377011(n,3))/2.

a(n) = (A002315(n) + 2*A008844(n) - 1)/2.

G.f.: (1 - 13*x - 41*x^2 + 21*x^3)/((1 - 34*x + x^2)*(1 - 6*x + x^2)*(1 - x)). - Andrew Howroyd, Oct 14 2024

Example

For n=2, the short leg is A002315(2) = 41 and the hypotenuse is A008844(n) = 841 so the semiperimeter is then a(2) = (41 + 840 + 841)/2 = 861.

Mathematica

s[n_]:=s[n]=Module[{a, b}, a=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2; b=(a^2-1)/2; {(a+2b+1)/2}]; semis={}; Do[semis=Join[semis, FullSimplify[s[n]]], {n, 0, 17}]; semis

Prog

(PARI) Vec((1 - 13*x - 41*x^2 + 21*x^3)/((1 - 34*x + x^2)*(1 - 6*x + x^2)*(1 - x)) + O(x^20)) \\ Andrew Howroyd, Oct 14 2024

Crossrefs

Cf. A377011, A002315, A078522, A008844.

Sequence in context: A214133 A119055 A324450 * A326279 A278195 A095656

Adjacent sequences: A377013 A377014 A377015 * A377017 A377018 A377019

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Oct 13 2024

Status

approved