%I #55 Jan 23 2019 18:48:26
%S 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,
%T 83,105,133,168,211,264,330,413,518,651,819,1030,1294,1624,2037,2555,
%U 3206,4025,5055,6349,7973,10010,12565,15771,19796,24851,31200,39173
%N Expansion of 1/(1 - x^7 - x^8 - ...).
%C A Lamé sequence of higher order.
%C a(n) = number of compositions of n in which each part is >= 7. - _Milan Janjic_, Jun 28 2010
%C 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
%C A017847(n) = |a(-n)| for n>=0. - _Michael Somos_, Oct 28 2018
%H Alois P. Heinz, <a href="/A017901/b017901.txt">Table of n, a(n) for n = 0..1000</a>
%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 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 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1).
%F G.f.: (x-1)/(x-1+x^7). - _Alois P. Heinz_, Aug 04 2008
%F 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
%F a(n) = A005709(n) - A005709(n-1). - _R. J. Mathar_, Sep 07 2016
%F 0 == a(n) + a(n+6) - a(n+7) for all n in Z. - _Michael Somos_, Oct 28 2018
%e 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
%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(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
%t LinearRecurrence[{1,0,0,0,0,0,1}, {1,0,0,0,0,0,0}, 60] (* _Jean-François Alcover_, Mar 28 2017 *)
%o (PARI) Vec((x-1)/(x-1+x^7)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o (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 */
%Y For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898, A017899, A017900, A017901, A017902, A017903, A017904.
%Y Cf. A005709, A017847.
%K nonn,easy
%O 0,15
%A _N. J. A. Sloane_
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