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 A017900 Expansion of 1/(1 - x^6 - x^7 - x^8 - ...). 5
 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 43, 55, 71, 92, 119, 153, 196, 251, 322, 414, 533, 686, 882, 1133, 1455, 1869, 2402, 3088, 3970, 5103, 6558, 8427, 10829, 13917, 17887, 22990, 29548, 37975, 48804, 62721, 80608 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS A Lamé sequence of higher order. Number of compositions of n into parts >= 6. - Milan Janjic, Jun 28 2010 a(n+6) equals the number of n-length binary words such that 0 appears only in a run which length is a multiple of 6. - Milan Janjic, Feb 17 2015 Same as sequence A005708 with 1, 0, 0, 0, 0, 0 prepended. - Linas Vepstas, Feb 06 2024 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5. J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380. Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5. Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1). FORMULA G.f.: 1/(1-Sum_{k>=6} x^k). G.f.: (1-x)/(1-x-x^6). - Alois P. Heinz, Aug 04 2008 For positive integers n and k such that k <= n <= 6*k, and 5 divides n-k, define c(n,k) = binomial(k,(n-k)/5), and c(n,k) = 0 otherwise. Then, for n >= 1, a(n+6) = Sum_{k=1..n} c(n,k). - Milan Janjic, Dec 09 2011 MAPLE f := proc(r) local t1, i; t1 := []; for i from 1 to r do t1 := [op(t1), 0]; od: for i from 1 to r+1 do t1 := [op(t1), 1]; od: for i from 2*r+2 to 50 do t1 := [op(t1), t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order a:= n-> (Matrix(6, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [1, 0\$4, 1][i], 0)))^n)[6, 6]: seq(a(n), n=0..80); # Alois P. Heinz, Aug 04 2008 MATHEMATICA f[n_] := If[n < 1, 1, Sum[ Binomial[ n - 5 k - 5, k], {k, 0, (n - 5)/6}]]; Array[f, 49, 0] (* Adi Dani, Robert G. Wilson v, Jul 04 2011 *) LinearRecurrence[{1, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, 60] (* Jean-François Alcover, Feb 13 2016 *) PROG (PARI) Vec((1-x)/(1-x-x^6)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012 CROSSREFS For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017904. Sequence in context: A328116 A193286 A098132 * A005708 A367637 A322853 Adjacent sequences: A017897 A017898 A017899 * A017901 A017902 A017903 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified August 8 12:42 EDT 2024. Contains 375021 sequences. (Running on oeis4.)