login
Expansion of (1-x)/(1-x-x^4).
20

%I #63 Jul 03 2020 14:55:05

%S 1,0,0,0,1,1,1,1,2,3,4,5,7,10,14,19,26,36,50,69,95,131,181,250,345,

%T 476,657,907,1252,1728,2385,3292,4544,6272,8657,11949,16493,22765,

%U 31422,43371,59864,82629,114051,157422,217286,299915,413966,571388,788674

%N Expansion of (1-x)/(1-x-x^4).

%C A Lamé sequence of higher order.

%C Essentially the same as A003269, which has much more information.

%C Number of compositions of n into parts >= 4. - _Joerg Arndt_, Aug 13 2012

%H Vincenzo Librandi, <a href="/A017898/b017898.txt">Table of n, a(n) for n = 0..1000</a>

%H Christian Ballot, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Ballot/ballot22.html">On Functions Expressible as Words on a Pair of Beatty Sequences</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.2.

%H I. M. Gessel, 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 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

%H T. Mansour, M. Shattuck, <a href="http://arxiv.org/abs/1410.6943">A monotonicity property for generalized Fibonacci sequences</a>, arXiv:1410.6943 [math.CO], 2014.

%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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1).

%F a(n) = a(n-1) + a(n-4). - _R. J. Mathar_, Mar 06 2008

%F G.f.: 1/(1-sum(k>=4, x^k)). - _Joerg Arndt_, Aug 13 2012

%F Apparently a(n) = hypergeometric([1-1/4*n, 5/4-1/4*n, 3/2-1/4*n, 7/4-1/4*n],[4/3-1/3*n, 5/3-1/3*n, 2-1/3*n], -4^4/3^3) for n>=13. - _Peter Luschny_, Sep 18 2014

%F a(n) = A003269(n+1)-A003269(n). - _R. J. Mathar_, Jun 10 2018

%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(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$2, 1][i] else 0 fi)^n)[4,4]: seq(a(n), n=0..42); # _Alois P. Heinz_, Aug 04 2008

%t LinearRecurrence[{1, 0, 0, 1}, {1, 0, 0, 0}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 11 2012 *)

%t CoefficientList[Series[(1-x)/(1-x-x^4),{x,0,50}],x] (* _Harvey P. Dale_, Sep 12 2019 *)

%o (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,0,1]^n*[1;0;0;0])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016

%Y For Lamé sequences of orders 1 through 9 see A000045, A000930, this one, and A017899-A017904.

%K nonn,easy

%O 0,9

%A _N. J. A. Sloane_