A154340 a(n) = ( (5 + 2*sqrt(2))^n - (5 - 2*sqrt(2))^n )/(4*sqrt(2)).

1, 10, 83, 660, 5189, 40670, 318487, 2493480, 19520521, 152816050, 1196311643, 9365243580, 73315137869, 573942237830, 4493065034527, 35173632302160, 275354217434641, 2155590425209690, 16874882555708003, 132103788328515300

Offset

1,2

Comments

First differences are in A164588.

Lim_{n -> infinity} a(n)/a(n-1) = 5 + 2*sqrt(2) = 7.8284271247....

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (10, -17).

Formula

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1, with a(0)=0, a(1)=1. - Philippe Deléham, Jan 12 2009

G.f.: x/(1 - 10*x + 17*x^2). - Klaus Brockhaus, Jan 12 2009, corrected Oct 08 2009

E.g.f.: (1/sqrt(8))*exp(5*x)*sinh(2*sqrt(2)*x). - G. C. Greubel, Sep 11 2016

Maple

A154340:=n->((5+2*sqrt(2))^n-(5-2*sqrt(2))^n)/(4*sqrt(2)): seq(simplify(A154340(n)), n=1..30); # Wesley Ivan Hurt, Sep 12 2016

Mathematica

Drop[CoefficientList[Series[x/(1-10*x+17*x^2), {x, 0, 30}], x], 1] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011, modified by G. C. Greubel, Jun 01 2019 *)
LinearRecurrence[{10, -17}, {1, 10}, 30] (* or *) Table[Simplify[((5 + 2*Sqrt[2])^n -(5-2*Sqrt[2])^n)/(4*Sqrt[2])], {n, 1, 30}] (* G. C. Greubel, Sep 11 2016 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+2*r)^n-(5-2*r)^n)/(4*r): n in [1..30] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009
(Sage) [lucas_number1(n, 10, 17) for n in range(1, 30)] # Zerinvary Lajos, Apr 26 2009
(Magma) I:=[1, 10]; [n le 2 select I[n] else 10*Self(n-1)-17*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 12 2016
(PARI) a(n)=([0, 1; -17, 10]^(n-1)*[1; 10])[1, 1] \\ Charles R Greathouse IV, Sep 12 2016

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A164588.

Sequence in context: A360698 A026954 A116879 * A037699 A037608 A055149

Adjacent sequences: A154337 A154338 A154339 * A154341 A154342 A154343

Keyword

nonn,easy

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 12 2009

Edited by Klaus Brockhaus, Oct 08 2009

Status

approved