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A015577 a(n+1) = 8*a(n) + 9*a(n-1), a(0) = 0, a(1) = 1.

%I A015577 #64 Sep 08 2022 08:44:40
%S A015577 0,1,8,73,656,5905,53144,478297,4304672,38742049,348678440,3138105961,
%T A015577 28242953648,254186582833,2287679245496,20589113209465,
%U A015577 185302018885184,1667718169966657,15009463529699912,135085171767299209,1215766545905692880,10941898913151235921
%N A015577 a(n+1) = 8*a(n) + 9*a(n-1), a(0) = 0, a(1) = 1.
%C A015577 Binomial transform is A011557, with a leading zero. - _Paul Barry_, Jul 09 2003
%C A015577 Number of walks of length n between any two distinct nodes of the complete graph K_10. Example: a(2) = 8 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJ are: ACB, ADB, AEB, AFB, AGB, AHB, AIB and AJB. - _Emeric Deutsch_, Apr 01 2004
%C A015577 The ratio a(n+1)/a(n) converges to 9 as n approaches infinity. - _Felix P. Muga II_, Mar 09 2014
%H A015577 Vincenzo Librandi, <a href="/A015577/b015577.txt">Table of n, a(n) for n = 0..1000</a>
%H A015577 Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, <a href="https://arxiv.org/abs/1911.01687">Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences</a>, arXiv:1911.01687 [math.CO], 2019.
%H A015577 Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A015577 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,9).
%F A015577 From _Paul Barry_, Jul 09 2003: (Start)
%F A015577 G.f.: x/((1+x)*(1-9*x)).
%F A015577 E.g.f. exp(4*x)*sinh(5*x)/5.
%F A015577 a(n) = (9^n - (-1)^n)/10. (End)
%F A015577 a(n) = 9^(n-1)-a(n-1). - _Emeric Deutsch_, Apr 01 2004
%F A015577 a(n) = round(9^n/10). - _Mircea Merca_, Dec 28 2010
%p A015577 seq(round(9^n/10),n=0..25); # _Mircea Merca_, Dec 28 2010
%t A015577 k=0;lst={k};Do[k=9^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008 *)
%t A015577 Table[(9^n - (-1)^n)/10, {n,0,30}] (* or *) LinearRecurrence[{8,9}, {0,1}, 30] (* _G. C. Greubel_, Jan 06 2018 *)
%o A015577 (PARI) A015577_vec(N=20)=Vec(O(x^N)+1/(1-8*x-9*x^2), -N-1) \\ _M. F. Hasler_, Jun 14 2008, edited Oct 25 2019
%o A015577 (PARI) for(n=0,30, print1((9^n - (-1)^n)/10, ", ")) \\ _G. C. Greubel_, Jan 06 2018
%o A015577 (PARI) apply( {A015577(n)=9^n\/10}, [0..25]) \\ _M. F. Hasler_, Oct 25 2019
%o A015577 (Sage) [lucas_number1(n,8,-9) for n in range(0, 19)] # _Zerinvary Lajos_, Apr 25 2009
%o A015577 (Magma) [Round(9^n/10): n in [0..30]]; // _Vincenzo Librandi_, Jun 24 2011
%o A015577 (Maxima)
%o A015577 a[0]:0$
%o A015577 a[n]:=9^(n-1)-a[n-1]$
%o A015577 A015577(n):=a[n]$
%o A015577 makelist(A015577(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */
%Y A015577 Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%K A015577 nonn,easy
%O A015577 0,3
%A A015577 _Olivier Gérard_
%E A015577 Extended by _T. D. Noe_, May 23 2011