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%I #84 Aug 03 2020 01:31:46
%S 1,0,0,0,0,1,1,1,1,1,2,3,4,5,6,8,11,15,20,26,34,45,60,80,106,140,185,
%T 245,325,431,571,756,1001,1326,1757,2328,3084,4085,5411,7168,9496,
%U 12580,16665,22076,29244,38740,51320,67985,90061,119305,158045,209365,277350
%N Expansion of 1/(1 -x^5 -x^6 -x^7 - ...).
%C a(n) is the number of compositions of n into parts >=5. - _Joerg Arndt_, Jun 22 2011
%C a(n+5) equals the number of binary words such that 0 appears only in runs whose lengths are a multiple of 5. - _Milan Janjic_, Feb 17 2015
%C a(n-5) equals the number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 2 or 3. - _Ethan Patrick White_, Jun 24 2020
%H Alois P. Heinz, <a href="/A017899/b017899.txt">Table of n, a(n) for n = 0..1000</a>
%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 J. D. Opdyke, <a href="http://dx.doi.org/10.1007/s10852-009-9116-2">A unified approach to algorithms generating unrestricted..</a>, J. Math. Model. Algor. 9 (2010) 53-97.
%H Ethan P. White, Richard K. Guy, Renate Scheidler, <a href="https://arxiv.org/abs/2006.15250">Difference Necklaces</a>, arXiv:2006.15250 [math.CO], 2020.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1).
%F G.f.: (1-x)/(1-x-x^5) = 1/(1-Sum_{k>=5} x^k).
%F For positive integers n and k such that k <= n <= 5*k, and 4 divides n-k, define c(n,k) = binomial(k,(n-k)/4), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+5) = 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(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$3, 1][i] else 0 fi)^n)[5,5]: seq(a(n), n=0..50); # _Alois P. Heinz_, Aug 04 2008
%t CoefficientList[ Series[(1 - x)/(1 - x - x^5), {x, 0, 50}], x] (* _Adi Dani_, Jun 25 2011 *)
%t LinearRecurrence[{1,0,0,0,1},{1,0,0,0,0},60] (* _Harvey P. Dale_, Jun 07 2015 *)
%o (PARI) Vec((1-x)/(1-x-x^5)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 21 2011
%Y For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017904.
%Y Apart from initial terms, same as A003520.
%K nonn,easy
%O 0,11
%A _N. J. A. Sloane_.
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