A015585 a(n) = 9*a(n-1) + 10*a(n-2).

0, 1, 9, 91, 909, 9091, 90909, 909091, 9090909, 90909091, 909090909, 9090909091, 90909090909, 909090909091, 9090909090909, 90909090909091, 909090909090909, 9090909090909091, 90909090909090909, 909090909090909091, 9090909090909090909, 90909090909090909091

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

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

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
• Index entries for linear recurrences with constant coefficients, signature (9,10).

Crossrefs

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

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

Cf. A094028, A098610, A098611.

Programs

Magma

[Round(10^n/11): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Mathematica

k=0;lst={k};Do[k=10^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
LinearRecurrence[{9,10},{0,1},30] (* Harvey P. Dale, Aug 08 2014 *)

PARI

a(n)=10^n\/11 \\ Charles R Greathouse IV, Jun 24 2011

Sage

[lucas_number1(n,9,-10) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

Sage

[abs(gaussian_binomial(n,1,-10)) for n in range(0,19)] # Zerinvary Lajos, May 28 2009

Formula

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

From Emeric Deutsch, Apr 01 2004: (Start)

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

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

From Henry Bottomley, Sep 17 2004: (Start)

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

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

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

E.g.f.: exp(-x)*(exp(11*x) - 1)/11. - Elmo R. Oliveira, Aug 17 2024

Extensions

Extended by T. D. Noe, May 23 2011