A153598 - OEIS

%I #19 Sep 08 2022 08:45:40

%S 1,14,150,1456,13484,121800,1084936,9586304,84301200,739246816,

%T 6471600224,56597049600,494665084096,4321846895744,37751262672000,

%U 329712720203776,2879419999940864,25145094869798400,219578008179897856,1917417750507843584

%N a(n) = ((7 + sqrt(3))^n - (7 - sqrt(3))^n)/(2*sqrt(3)).

%C lim_{n -> infinity} a(n)/a(n-1) = 7 + sqrt(3) = 8.73205080756887729....

%H G. C. Greubel, <a href="/A153598/b153598.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-46).

%F G.f.: x/(1 - 14*x + 46*x^2). - _Klaus Brockhaus_, Dec 31 2008, (corrected Oct 11 2009)

%F a(n) = 14*a(n-1) - 46*a(n-2) for n>1; a(0)=0, a(1)=1. - _Philippe Deléham_, Jan 01 2009

%t Join[{a=1,b=14},Table[c=14*b-46*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)

%t LinearRecurrence[{14,-46},{1,14},20] (* _Harvey P. Dale_, Dec 05 2015 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((7+r)^n-(7-r)^n)/(2*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Dec 31 2008

%o (Magma) I:=[1,14]; [n le 2 select I[n] else 14*Self(n-1)-46*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Aug 23 2016

%Y Cf. A002194 (decimal expansion of sqrt(3)).

%K nonn

%O 1,2

%A Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

%E Typo corrected and extended beyond a(7) by _Klaus Brockhaus_, Dec 31 2008

%E Edited by _Klaus Brockhaus_, Oct 11 2009