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A107066
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Expansion of 1/(1-2*x+x^5).
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15
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1, 2, 4, 8, 16, 31, 60, 116, 224, 432, 833, 1606, 3096, 5968, 11504, 22175, 42744, 82392, 158816, 306128, 590081, 1137418, 2192444, 4226072, 8146016, 15701951, 30266484, 58340524, 112454976, 216763936, 417825921, 805385358, 1552430192, 2992405408, 5768046880
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OFFSET
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0,2
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COMMENTS
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Row sums of number triangle A107065.
a(n) is the number of binary words of length n containing no subword 01011. - Alois P. Heinz, Mar 14 2012
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LINKS
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FORMULA
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a(n) = 2*a(n-1) - a(n-5).
a(n) = Sum_{k=0..floor(n/5)} C(n-4*k, k) * 2^(n-2*k) *(-1)^k.
G.f.: 1/((1 - x)*(1 - x - x^2 - x^3 - x^4)).
Setting k = 1 in the double recurrence for array A140996, we get that a(n+5) = 1 + a(n+1) + a(n+2) + a(n+3) + a(n+4) for n >= 0, which of course we can prove using other methods as well. See also Dunkel (1925).
(End)
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EXAMPLE
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G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 31*x^5 + 60*x^6 + 116*x^7 + 224*x^8 + ...
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MATHEMATICA
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LinearRecurrence[{2, 0, 0, 0, -1}, {1, 2, 4, 8, 16}, 40] (* G. C. Greubel, Jun 12 2019 *)
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PROG
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(PARI) {a(n) = if( n<0, n = -n; polcoeff( -x^5 / (1 - 2*x^4 + x^5) + x * O(x^n), n), polcoeff( 1 / (1 - 2*x + x^5) + x * O(x^n), n))} /* Michael Somos, Dec 28 2012 */
(Magma) I:=[1, 2, 4, 8, 16]; [n le 5 select I[n] else 2*Self(n-1) - Self(n-5): n in [1..40]]; // G. C. Greubel, Jun 12 2019
(Sage) (1/(1-2*x+x^5)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019
(GAP) a:=[1, 2, 4, 8, 16];; for n in [6..40] do a[n]:=2*a[n-1]-a[n-5]; od; a; # G. C. Greubel, Jun 12 2019
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CROSSREFS
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Column k = 1 of array A140996 (with a different offset) and second main diagonal of A140995.
Column k = 4 of A172119 (with a different offset).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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