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A017902
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Expansion of 1/(1 - x^8 - x^9 - ...).
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4
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1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078
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OFFSET
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0,17
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COMMENTS
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A Lamé sequence of higher order.
a(n) = number of compositions of n in which each part is >=8. - Milan Janjic, Jun 28 2010
a(n+8) equals the number of n-length binary words such that 0 appears only in a run which length is a multiple of 8. - Milan Janjic, Feb 17 2015
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LINKS
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FORMULA
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For positive integers n and k such that k <= n <= 8*k, and 7 divides n-k, define c(n,k) = binomial(k,(n-k)/7), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+8) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
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MAPLE
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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(8, (i, j)-> if (i=j-1) then 1 elif j=1 then [1, 0$6, 1][i] else 0 fi)^n)[8, 8]: seq(a(n), n=0..53); # Alois P. Heinz, Aug 04 2008
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MATHEMATICA
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LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0}, 60] (* Jean-François Alcover, Feb 13 2016 *)
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PROG
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(PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 1; 1, 0, 0, 0, 0, 0, 0, 1]^n*[1; 0; 0; 0; 0; 0; 0; 0])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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