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%I #70 Feb 07 2024 12:58:40
%S 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,
%T 153,196,251,322,414,533,686,882,1133,1455,1869,2402,3088,3970,5103,
%U 6558,8427,10829,13917,17887,22990,29548,37975,48804,62721,80608
%N Expansion of 1/(1 - x^6 - x^7 - x^8 - ...).
%C A Lamé sequence of higher order.
%C Number of compositions of n into parts >= 6. - _Milan Janjic_, Jun 28 2010
%C 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
%C Same as sequence A005708 with 1, 0, 0, 0, 0, 0 prepended. - _Linas Vepstas_, Feb 06 2024
%H Alois P. Heinz, <a href="/A017900/b017900.txt">Table of n, a(n) for n = 0..1000</a>
%H I. M. Gessel and Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5.
%H J. Hermes, <a href="http://dx.doi.org/10.1007/BF01446684">Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden</a>, Math. Ann., 45 (1894), 371-380.
%H Augustine O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Munagi/munagi10.html">Integer Compositions and Higher-Order Conjugation</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1).
%F G.f.: 1/(1-Sum_{k>=6} x^k).
%F G.f.: (1-x)/(1-x-x^6). - _Alois P. Heinz_, Aug 04 2008
%F 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
%p 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
%p 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
%t 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 *)
%t LinearRecurrence[{1, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0}, 60] (* _Jean-François Alcover_, Feb 13 2016 *)
%o (PARI) Vec((1-x)/(1-x-x^6)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017904.
%K nonn,easy
%O 0,13
%A _N. J. A. Sloane_
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