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%I #41 Sep 08 2022 08:45:55
%S 1,7,47,315,2111,14147,94807,635355,4257871,28534387,191224967,
%T 1281505995,8588092031,57553632227,385699241527,2584787426235,
%U 17322113500591,116085219651667,777952310560487,5213495734620075,34938565521219551
%N Expansion of 1/(1 - 7*x + 2*x^2).
%C The first differences are in A122074.
%C a(n+1) equals the number of words of length n over {0,1,2,3,4,5,6} avoiding 01 and 02. - _Milan Janjic_, Dec 17 2015
%H Bruno Berselli, <a href="/A186446/b186446.txt">Table of n, a(n) for n = 0..800</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.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-2).
%F G.f.: 1/(1-7*x+2*x^2).
%F a(n) = ((7+sqrt(41))^(1+n)-(7-sqrt(41))^(1+n))/(2^(1+n)*sqrt(41)).
%F a(n) = 7*a(n-1)-2*a(n-2), with a(0)=1, a(1)=7.
%t CoefficientList[Series[1 / (1 - 7 x + 2 x^2), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 19 2013 *)
%t LinearRecurrence[{7,-2},{1,7},30] (* _Harvey P. Dale_, Aug 06 2017 *)
%o (Magma) m:=21; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-7*x+2*x^2)));
%o (Magma) I:=[1,7]; [n le 2 select I[n] else 7*Self(n-1)-2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 19 2013
%o (PARI) Vec(1/(1-7*x+2*x^2) + O(x^100)) \\ _Altug Alkan_, Dec 17 2015
%Y For similar closed formulas: A015446 [((1+sqrt(41))^(1+n)-(1-sqrt(41))^(1+n))/(2^(1+n)*sqrt(41))], A015525 [((3+sqrt(41))^n-(3-sqrt(41))^n)/(2^n*sqrt(41))], A015537 [((5+sqrt(41))^n-(5-sqrt(41))^n)/(2^n*sqrt(41))], A178869 [((9+sqrt(41))^n-(9-sqrt(41))^n)/(2^n*sqrt(41))].
%Y Same recurrence as in A122074, A003771.
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Feb 21 2011
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