OFFSET
0,3
COMMENTS
Binomial transform is A011557, with a leading zero. - Paul Barry, Jul 09 2003
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
The ratio a(n+1)/a(n) converges to 9 as n approaches infinity. - Felix P. Muga II, Mar 09 2014
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.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (8,9).
FORMULA
From Paul Barry, Jul 09 2003: (Start)
G.f.: x/((1+x)*(1-9*x)).
E.g.f. exp(4*x)*sinh(5*x)/5.
a(n) = (9^n - (-1)^n)/10. (End)
a(n) = 9^(n-1)-a(n-1). - Emeric Deutsch, Apr 01 2004
a(n) = round(9^n/10). - Mircea Merca, Dec 28 2010
MAPLE
seq(round(9^n/10), n=0..25); # Mircea Merca, Dec 28 2010
MATHEMATICA
k=0; lst={k}; Do[k=9^n-k; AppendTo[lst, k], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[(9^n - (-1)^n)/10, {n, 0, 30}] (* or *) LinearRecurrence[{8, 9}, {0, 1}, 30] (* G. C. Greubel, Jan 06 2018 *)
PROG
(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
(PARI) for(n=0, 30, print1((9^n - (-1)^n)/10, ", ")) \\ G. C. Greubel, Jan 06 2018
(PARI) apply( {A015577(n)=9^n\/10}, [0..25]) \\ M. F. Hasler, Oct 25 2019
(Sage) [lucas_number1(n, 8, -9) for n in range(0, 19)] # Zerinvary Lajos, Apr 25 2009
(Magma) [Round(9^n/10): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011
(Maxima)
a[0]:0$
a[n]:=9^(n-1)-a[n-1]$
A015577(n):=a[n]$
makelist(A015577(n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Extended by T. D. Noe, May 23 2011
STATUS
approved