A154248 a(n) = ( (7 + sqrt(7))^n - (7 - sqrt(7))^n )/(2*sqrt(7)).

1, 14, 154, 1568, 15484, 150920, 1462552, 14137088, 136492048, 1317130976, 12707167648, 122580846080, 1182430803904, 11405635719296, 110016806306176, 1061198588076032, 10236074368205056, 98734700455677440

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

Lim_{n -> infinity} a(n)/a(n-1) = 7 + sqrt(7) = 9.6457513110....

Links

• G. C. Greubel, Table of n, a(n) for n = 1..750
• Index entries for linear recurrences with constant coefficients, signature (14,-42).

Crossrefs

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

Programs

Magma

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

Maple

a:= n-> (<<0|1>, <-42|14>>^n)[1, 2]:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 22 2013

Mathematica

Join[{a=1,b=14},Table[c=14*b-42*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
LinearRecurrence[{14,-42},{1,14},25] (* or *) Table[( (7 + sqrt(7))^n - (7 - 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) = 14*a(n-1) - 42*a(n-2) for n>1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 14x + 42x^2). (End)

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

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009