%I #42 Dec 30 2023 23:41:25

%S 0,1,7,51,371,2699,19635,142843,1039171,7559883,54997523,400102427,

%T 2910712035,21175189099,154047747763,1120684612539,8152887783299,

%U 59311583708171,431486861523795,3139031198082907,22836192109627939

%N Expansion of x/(1 - 7*x - 2*x^2).

%C For n>0, a(n) equals the number of words of length n-1 over {0,1,...,8} in which 0 and 1 avoid runs of odd lengths. - _Milan Janjic_, Jan 08 2017

%H Vincenzo Librandi, <a href="/A015555/b015555.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,2).

%F a(n) = 7*a(n-1) + 2*a(n-2).

%F E.g.f.: (exp(x*(7 + sqrt(57))/2) - exp(x*(7 - sqrt(57))/2))/sqrt(57). - _Iain Fox_, Dec 30 2017

%t Join[{a=0,b=1},Table[c=7*b+2*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011 *)

%t LinearRecurrence[{7, 2}, {0, 1}, 30] (* _Vincenzo Librandi_ Nov 13 2012 *)

%o (Sage) [lucas_number1(n,7,-2) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 24 2009

%o (Magma) [n le 2 select n-1 else 7*Self(n-1) + 2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 13 2012

%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1-7*x-2*x^2))) \\ _G. C. Greubel_, Dec 30 2017

%K nonn,easy

%O 0,3

%A _Olivier Gérard_