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URL: https://oeis.org/A017898

⇱ A017898 - OEIS

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A017898
Expansion of (1-x)/(1-x-x^4).
20
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
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
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
KEYWORD
nonn,easy
STATUS
approved