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

1,2

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).

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

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

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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
(PARI) a(n)=([0, 1; -41, 14]^(n-1)*[1; 14])[1, 1] \\ Charles R Greathouse IV, May 30 2026

Crossrefs

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

Sequence in context: A006865 A391779 A263474 * A001707 A078999 A016157

Adjacent sequences: A154344 A154345 A154346 * A154348 A154349 A154350

Keyword

nonn,easy,changed

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