%I #48 Aug 17 2024 20:38:09

%S 0,1,9,91,909,9091,90909,909091,9090909,90909091,909090909,9090909091,

%T 90909090909,909090909091,9090909090909,90909090909091,

%U 909090909090909,9090909090909091,90909090909090909,909090909090909091,9090909090909090909,90909090909090909091

%N a(n) = 9*a(n-1) + 10*a(n-2).

%C Number of walks of length n between any two distinct nodes of the complete graph K_11. Example: a(2)=9 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJK are: ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB and AKB. - _Emeric Deutsch_, Apr 01 2004

%C Beginning with n=1 and a(1)=1, these are the positive integers whose balanced base-10 representations (A097150) are the first n digits of 1,-1,1,-1,.... Also, a(n) = (-1)^(n-1)*A014992(n) = |A014992(n)| for n >= 1. - _Rick L. Shepherd_, Jul 30 2004

%C General form: k=10^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008

%H Vincenzo Librandi, <a href="/A015585/b015585.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,10).

%F a(n) = 9*a(n-1) + 10*a(n-2).

%F From _Emeric Deutsch_, Apr 01 2004: (Start)

%F a(n) = 10^(n-1) - a(n-1).

%F G.f.: x/(1 - 9x - 10x^2). (End)

%F From _Henry Bottomley_, Sep 17 2004: (Start)

%F a(n) = round(10^n/11).

%F a(n) = (10^n - (-1)^n)/11.

%F a(n) = A098611(n)/11 = 9*A094028(n+1)/A098610(n). (End)

%F E.g.f.: exp(-x)*(exp(11*x) - 1)/11. - _Elmo R. Oliveira_, Aug 17 2024

%t k=0;lst={k};Do[k=10^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008 *)

%t LinearRecurrence[{9,10},{0,1},30] (* _Harvey P. Dale_, Aug 08 2014 *)

%o (Sage) [lucas_number1(n,9,-10) for n in range(0, 19)] # _Zerinvary Lajos_, Apr 26 2009

%o (Sage) [abs(gaussian_binomial(n,1,-10)) for n in range(0,19)] # _Zerinvary Lajos_, May 28 2009

%o (Magma) [Round(10^n/11): n in [0..30]]; // _Vincenzo Librandi_, Jun 24 2011

%o (PARI) a(n)=10^n\/11 \\ _Charles R Greathouse IV_, Jun 24 2011

%Y Cf. A014992 (q-integers for q=-10), A097150.

%Y Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008

%Y Cf. A094028, A098610, A098611.

%K nonn,easy

%O 0,3

%A _Olivier Gérard_

%E Extended by _T. D. Noe_, May 23 2011