A292034 Linear divisibility sequence based on Salem number of order 4 (case t=6, see formula).

1, 6, 29, 144, 725, 3654, 18409, 92736, 467161, 2353350, 11855141, 59720976, 300847949, 1515539334, 7634619025, 38459844864, 193743743089, 975995564166, 4916635376621, 24767841488400, 124769466312581, 628533565640646, 3166275009522169, 15950297619676224, 80350567588455625

Offset

1,2

Links

Table of n, a(n) for n=1..25.

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Linear divisibility sequences and Salem numbers, arXiv:1709.01995 [math.NT], 2017.

Formula

a(n) = round(lambda(6)*alpha(6)^n)

where alpha(t) = (t+sqrt((t-4)*t+8)+sqrt(2)*sqrt(t*(t+sqrt((t-4)*t+8)-2)-4))/4

and lambda(t) = 1/sqrt((t-4)*t+8).

Conjectures from Colin Barker, Dec 17 2017: (Start)

G.f.: x*(1 - x)*(1 + x) / (1 - 6*x + 6*x^2 - 6*x^3 + x^4).

a(n) = 6*a(n-1) - 6*a(n-2) + 6*a(n-3) - a(n-4) for n>4.

(End)

Mathematica

alpha[t_] := (t + Sqrt[(t - 4) t + 8] + Sqrt[2] Sqrt[t (t + Sqrt[(t - 4) t + 8] - 2) - 4])/4;
lambda[t_] := 1/Sqrt[(t - 4) t + 8];
a[n_] := Round[lambda[6] alpha[6]^n] ;
Array[a, 25] (* Jean-François Alcover, Feb 02 2019 *)

Prog

(PARI) alpha(t) = (t+sqrt((t-4)*t+8)+sqrt(2)*sqrt(t*(t+sqrt((t-4)*t+8)-2)-4))/4;
lambda(t) = 1/sqrt((t-4)*t+8);
a(n) = my(ca=alpha(6), cl=lambda(6)); round(cl*ca^n);

Crossrefs

Cf. A292035.

Sequence in context: A012325 A125785 A186651 * A108982 A059724 A000708

Adjacent sequences: A292031 A292032 A292033 * A292035 A292036 A292037

Keyword

nonn

Author

Michel Marcus, Sep 08 2017

Status

approved