A156570 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=17, a(2)=65.

17, 65, 373, 2173, 12665, 73817, 430237, 2507605, 14615393, 85184753, 496493125, 2893773997, 16866150857, 98303131145, 572952636013, 3339412684933, 19463523473585, 113441728156577, 661186845465877, 3853679344638685

Offset: 1

Author

Klaus Brockhaus, Feb 11 2009, Feb 16 2009

Keywords

• nonn

Comments

lim_{n -> infinity} a(n)/a(n-1) = 3+2*sqrt(2).

Links

• Index entries for linear recurrences with constant coefficients, signature (6, -1).

Crossrefs

First trisection of A156567.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156568, A156569.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((74+47*r2)*(3-2*r2)^n+(74-47*r2)*(3+2*r2)^n)/4: n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ];

PARI

{m=20; v=concat([17, 65], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

Formula

a(n) = ((74+47*sqrt(2))*(3-2*sqrt(2))^n+(74-47*sqrt(2))*(3+2*sqrt(2))^n)/4.

G.f.: x*(17-37*x)/(1-6*x+x^2).

Extensions

G.f. corrected by Klaus Brockhaus, Sep 22 2009