OFFSET
0,3
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
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).
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.
E.g.f.: exp(-x)*(exp(11*x) - 1)/11. - Elmo R. Oliveira, Aug 17 2024
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 *)
PROG
(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
(Magma) [Round(10^n/11): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011
(PARI) a(n)=10^n\/11 \\ Charles R Greathouse IV, Jun 24 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Extended by T. D. Noe, May 23 2011
STATUS
approved