A173127 a(n) = sinh((2n-1)*arcsinh(3)).

-3, 3, 117, 4443, 168717, 6406803, 243289797, 9238605483, 350823718557, 13322062699683, 505887558869397, 19210405174337403, 729489509065951917, 27701390939331835443, 1051923366185543794917, 39945386524111332371403

Offset: 0

Author

Artur Jasinski, Feb 10 2010

Keywords

• sign
• easy

Comments

Numbers n such that ((n^2 + 1)/10) is a square. - Vincenzo Librandi, Jan 02 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (38,-1).

Crossrefs

Cf. A058331 A001079, A037270, A071253, A108741, A132592, A146311-A146313, A173115, A173116, A173121.

Programs

Magma

[-3] cat [n: n in [0..10^7]|IsSquare((n^2+1)/10)]; // Vincenzo Librandi, Jan 02 2012

Mathematica

LinearRecurrence[{38,-1},{-3,3},30] (* Harvey P. Dale, Jan 14 2015 *)

Python

from itertools import islice
def A173127_gen(): # generator of terms
    x, y = -30, 10
    while True:
        yield x//10
        x, y = x*19+y*60, x*6+y*19
A173127_list = list(islice(A173127_gen(),20)) # Chai Wah Wu, Apr 24 2025

Formula

a(n) = (1/2)*((-3+sqrt(10))*(19+6*sqrt(10))^n + (-3-sqrt(10))*(19-6*sqrt(10))^n).

a(n) = -a(-n+1).

G.f.: -3*(1-39*x)/(1-38*x+x^2). - Bruno Berselli, Jan 03 2011

E.g.f.: exp(19*x)*(-3*cosh(6*sqrt(10)*x) + sqrt(10)*sinh(6*sqrt(10)*x)). - Stefano Spezia, Apr 24 2025