OFFSET
0,9
COMMENTS
a(n+7) equals the number of binary words of length n having at least 8 zeros between every two successive ones. - Milan Janjic, Feb 09 2015
a(n) is the number of compositions of n+1 into parts 1 and 9. - Joerg Arndt, May 19 2018
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
D. Kleitman, Solution to Problem E3274, Amer. Math. Monthly, 98 (1991), 958-959.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
D. Newman, Problem E3274, Amer. Math. Monthly, 95 (1988), 555.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 382
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 1).
FORMULA
G.f.: (1+x^8)/(1-x-x^9).
For positive integers n and k such that k <= n <= 9*k, and 8 divides n-k, define c(n,k) = binomial(k,(n-k)/8), and c(n,k) = 0, otherwise. Then, for n>= 1, a(n-1) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011
MAPLE
A005711:=-(1+z**8)/(-1+z+z**9); # Simon Plouffe in his 1992 dissertation
ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, b))), X = Sequence(b, card >= 8)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=9..65); # Zerinvary Lajos, Mar 26 2008
M:= Matrix(9, (i, j)-> if j=1 and member(i, [1, 9]) then 1 elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n+1))[1, 1]; seq(a(n), n=0..60); # Alois P. Heinz, Jul 27 2008
MATHEMATICA
CoefficientList[Series[(1+x^8)/(1-x-x^9), {x, 0, 57}], x] (* Michael De Vlieger, May 20 2018 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 2}, 60] (* Harvey P. Dale, Jul 30 2022 *)
PROG
(PARI) x='x+O('x^66); Vec((1+x^8)/(1-x-x^9)) /* Joerg Arndt, Jun 25 2011 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved