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A017899 Expansion of 1/(1 -x^5 -x^6 -x^7 - ...). 6
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, 245, 325, 431, 571, 756, 1001, 1326, 1757, 2328, 3084, 4085, 5411, 7168, 9496, 12580, 16665, 22076, 29244, 38740, 51320, 67985, 90061, 119305, 158045, 209365, 277350 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

a(n) is the number of compositions of n into parts >=5. - Joerg Arndt, Jun 22 2011

a(n+5) equals the number of binary words such that 0 appears only in a run which length is a multiple of 5. - Milan Janjic, Feb 17 2015

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.

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 (1,0,0,0,1).

FORMULA

G.f.: (1-x)/(1-x-x^5) = 1/(1-sum(k>=5, x^k)).

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(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011

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(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

MATHEMATICA

CoefficientList[ Series[(1 - x)/(1 - x - x^5), {x, 0, 50}], x] (* Adi Dani, Jun 25 2011 *)

LinearRecurrence[{1, 0, 0, 0, 1}, {1, 0, 0, 0, 0}, 60] (* Harvey P. Dale, Jun 07 2015 *)

PROG

(PARI) Vec((1-x)/(1-x-x^5)+O(x^99)) \\ Charles R Greathouse IV, Jun 21 2011

CROSSREFS

For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898-A017904.

Apart from initial terms, same as A003520.

Sequence in context: A218930 A026483 A098131 * A003520 A101915 A295073

Adjacent sequences:  A017896 A017897 A017898 * A017900 A017901 A017902

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified September 25 22:57 EDT 2018. Contains 315425 sequences. (Running on oeis4.)