A017898 Expansion of (1-x)/(1-x-x^4).

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, 476, 657, 907, 1252, 1728, 2385, 3292, 4544, 6272, 8657, 11949, 16493, 22765, 31422, 43371, 59864, 82629, 114051, 157422, 217286, 299915, 413966, 571388, 788674

Offset

0,9

Comments

A Lamé sequence of higher order.

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

Number of compositions of n into parts >= 4. - Joerg Arndt, Aug 13 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Christian Ballot, On Functions Expressible as Words on a Pair of Beatty Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.2.

I. M. Gessel, 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.

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

T. Mansour, M. Shattuck, A monotonicity property for generalized Fibonacci sequences, arXiv:1410.6943 [math.CO], 2014.

Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.

J. D. Opdyke, A unified approach to algorithms generating unrestricted.., J. Math. Model. Algor. 9 (2010) 53-97

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Formula

a(n) = a(n-1) + a(n-4). - R. J. Mathar, Mar 06 2008

G.f.: 1/(1-sum(k>=4, x^k)). - Joerg Arndt, Aug 13 2012

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

a(n) = A003269(n+1)-A003269(n). - R. J. Mathar, Jun 10 2018

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(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

Mathematica

LinearRecurrence[{1, 0, 0, 1}, {1, 0, 0, 0}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
CoefficientList[Series[(1-x)/(1-x-x^4), {x, 0, 50}], x] (* Harvey P. Dale, Sep 12 2019 *)

Prog

(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

Crossrefs

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

Sequence in context: A238874 A099559 A247084 * A003269 A367794 A352043

Adjacent sequences: A017895 A017896 A017897 * A017899 A017900 A017901

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved