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A017901 Expansion of 1/(1 - x^7 - x^8 - ...). 2
1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 53, 66, 83, 105, 133, 168, 211, 264, 330, 413, 518, 651, 819, 1030, 1294, 1624, 2037, 2555, 3206, 4025, 5055, 6349, 7973, 10010, 12565, 15771, 19796, 24851, 31200, 39173 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,15

COMMENTS

A Lamé sequence of higher order.

a(n) = number of compositions of n in which each part is >= 7. - Milan Janjic, Jun 28 2010

a(n+7) equals the number of n-length binary words such that 0 appears only in a run length that is a multiple of 7. - Milan Janjic, Feb 17 2015

A017847(n) = |a(-n)| for n>=0. - Michael Somos, Oct 28 2018

LINKS

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

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.

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

FORMULA

G.f.: (x-1)/(x-1+x^7). - Alois P. Heinz, Aug 04 2008

For positive integers n and k such that k <= n <= 7*k, and 6 divides n-k, define c(n,k) = binomial(k,(n-k)/6), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+7) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011

a(n) = A005709(n) - A005709(n-1). - R. J. Mathar, Sep 07 2016

0 == a(n) + a(n+6) - a(n+7) for all n in Z. - Michael Somos, Oct 28 2018

EXAMPLE

G.f. = 1 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + 2*x^14 + ... - Michael Somos, Oct 28 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(7, (i, j)-> if (i=j-1) then 1 elif j=1 then [1, 0$5, 1][i] else 0 fi)^n)[7, 7]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 04 2008

MATHEMATICA

LinearRecurrence[{1, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0}, 60] (* Jean-François Alcover, Mar 28 2017 *)

PROG

(PARI) Vec((x-1)/(x-1+x^7)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

(PARI) {a(n) = if( n < 0, polcoeff( 1 / (1 + x^6 - x^7) + x * O(x^-n), -n), polcoeff( (1 - x) / (1 - x - x^7) + x * O(x^n), n))}; /* Michael Somos, Oct 28 2018 */

CROSSREFS

For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898, A017899, A017900, A017901, A017902, A017903, A017904.

Cf. A005709, A017847.

Sequence in context: A215775 A236310 A005709 * A101917 A322854 A322802

Adjacent sequences:  A017898 A017899 A017900 * A017902 A017903 A017904

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 18 04:47 EST 2019. Contains 319269 sequences. (Running on oeis4.)