%I #44 Sep 08 2022 08:45:40
%S 1,6,34,192,1084,6120,34552,195072,1101328,6217824,35104288,198190080,
%T 1118931904,6317211264,35665403776,201358000128,1136817193216,
%U 6418187159040,36235488567808,204576557088768,1154988365396992
%N a(n) = 6*a(n-1) - 2*a(n-2) for n>1; a(1)=1, a(2)=6.
%C Binomial transform of A126473.
%C lim_{n -> infinity} a(n)/a(n-1) = 3+sqrt(7) = 5.6457513110....
%C a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} avoiding 01 and 02. - _Milan Janjic_, Dec 17 2015
%H Vincenzo Librandi, <a href="/A154244/b154244.txt">Table of n, a(n) for n = 1..500</a>
%H Tomislav Doslic, <a href="http://dx.doi.org/10.1007/s10910-013-0167-2">Planar polycyclic graphs and their Tutte polynomials</a>, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. See Cor. 3.7(e).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-2).
%F a(n) = ((3 + sqrt(7))^n - (3 - sqrt(7))^n)/(2*sqrt(7)).
%F G.f.: x/(1-6*x+2*x^2). - _Philippe Deléham_, Jan 06 2009
%t a[n_]:=(MatrixPower[{{1,3},{1,5}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)
%t LinearRecurrence[{6, -2}, {1, 6}, 40] (* _Vincenzo Librandi_, Feb 02 2012 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-7); S:=[ ((3+r)^n-(3-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 07 2009
%o (Magma) I:=[1, 6]; [n le 2 select I[n] else 6*Self(n-1)-2*Self(n-2): n in [1..50]]; // _Vincenzo Librandi_, Feb 02 2012
%o (Sage) [lucas_number1(n,6,2) for n in range(1, 22)] # _Zerinvary Lajos_, Apr 22 2009
%o (Maxima) a[1]:1$ a[2]:6$ a[n]:=6*a[n-1]-2*a[n-2]$ makelist(a[n], n, 1, 21); // _Bruno Berselli_, May 30 2011
%o (PARI) Vec(1/(1-6*x+2*x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Dec 28 2011
%Y Equals 1 followed by 2*A010913 (Pisot sequence E(3,17)).
%Y Cf. A010465 (decimal expansion of square root of 7), A126473.
%K nonn,easy
%O 1,2
%A Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
%E Extended beyond a(7) by _Klaus Brockhaus_, Jan 07 2009
%E Edited by _Klaus Brockhaus_, Oct 06 2009
%E Name (corrected) from _Philippe Deléham_, Jan 06 2009
|