A154249 a(n) = ( (8 + sqrt(7))^n - (8 - sqrt(7))^n )/(2*sqrt(7)).

1, 16, 199, 2272, 25009, 270640, 2904727, 31049152, 331216993, 3529670224, 37595354983, 400334476960, 4262416397329, 45379597170544, 483115820080951, 5143216082574208, 54753855576573121, 582898372518440080

Offset: 1

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Keywords

• nonn

Comments

lim_{n -> infinity} a(n)/a(n-1) = 8 + sqrt(7) = 10.6457513110....

Links

• G. C. Greubel, Table of n, a(n) for n = 1..750
• Index entries for linear recurrences with constant coefficients, signature (16, -57).

Crossrefs

Cf. A010465 (decimal expansion of square root of 7).

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((8+r)^n-(8-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

Maple

seq(expand((8+sqrt(7))^n-(8-sqrt(7))^n)/sqrt(28), n = 1 .. 20); # Emeric Deutsch, Jan 08 2009

Mathematica

Join[{a=1,b=16},Table[c=16*b-57*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
LinearRecurrence[{16,-57},{1,16},25] (* or *) Table[( (8 + sqrt(7))^n - (8 - sqrt(7))^n )/(2*sqrt(7)), {n,1,25}] (* G. C. Greubel, Sep 08 2016 *)

Formula

From Philippe Deléham, Jan 06 2009: (Start)

a(n) = 16*a(n-1)-57*a(n-2) for n>1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 16*x + 57*x^2). (End)

E.g.f.: (1/sqrt(7))*exp(8*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 08 2016

Extensions

Extended by Emeric Deutsch and Klaus Brockhaus, Jan 08 2009

Edited by Klaus Brockhaus, Oct 06 2009