A154347 a(n) = ( (7 + 2*sqrt(2))^n - (7 - 2*sqrt(2))^n )/(4*sqrt(2)).

1, 14, 155, 1596, 15989, 158410, 1562191, 15375864, 151212265, 1486561286, 14612155139, 143621159220, 1411597868381, 13873902629314, 136359124206775, 1340197731092976, 13172044142823889, 129460511024722430

Offset: 1

Author

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

Keywords

• nonn

Comments

Fourth binomial transform of A054489.

lim_{n -> infinity} a(n)/a(n-1) = 7 + 2*sqrt(2) = 9.8284271247....

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (14,-41).

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A054489.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((7+2*r)^n-(7-2*r)^n)/(4*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009

Maple

A154347:=n->((7+2*sqrt(2))^n-(7-2*sqrt(2))^n)/(4*sqrt(2)): seq(simplify(A154347(n)), n=1..30); # Wesley Ivan Hurt, Sep 13 2016

Mathematica

LinearRecurrence[{14,-41},{1,14}, 25] (* or *) Table[( (7 + 2*sqrt(2))^n - (7 - 2*sqrt(2))^n )/(4*sqrt(2)),{n,1,25}] (* G. C. Greubel, Sep 13 2016 *)

Formula

a(n) = 14*a(n-1) - 41*a(n-2) for n>1, with a(0)=0, a(1)=1. - Philippe Deléham, Jan 12 2009

G.f.: x/(1 - 14*x + 41*x^2). - Klaus Brockhaus, Jan 12 2009, corrected Oct 08 2009

E.g.f.: (1/4*sqrt(2))*exp(7*x)*sinh(2*sqrt(2)*x). - G. C. Greubel, Sep 13 2016

Extensions

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

Edited by Klaus Brockhaus, Oct 08 2009