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%I #60 Sep 08 2022 08:45:18
%S 1,2,4,8,16,31,60,116,224,432,833,1606,3096,5968,11504,22175,42744,
%T 82392,158816,306128,590081,1137418,2192444,4226072,8146016,15701951,
%U 30266484,58340524,112454976,216763936,417825921,805385358,1552430192,2992405408,5768046880
%N Expansion of 1/(1-2*x+x^5).
%C Row sums of number triangle A107065.
%C Same as A018922 plus first 3 additional terms. - _Vladimir Joseph Stephan Orlovsky_, Jul 08 2011
%C a(n) is the number of binary words of length n containing no subword 01011. - _Alois P. Heinz_, Mar 14 2012
%H Vincenzo Librandi, <a href="/A107066/b107066.txt">Table of n, a(n) for n = 0..1000</a>
%H O. Dunkel, <a href="http://www.jstor.org/stable/2298801">Solutions of a probability difference equation</a>, Amer. Math. Monthly, 32 (1925), 354-370; see p. 356.
%H T. Langley, J. Liese, and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order</a>, J. Int. Seq. 14 (2011), Article #11.4.2.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,0,-1).
%F a(n) = 2*a(n-1) - a(n-5).
%F a(n) = Sum_{k=0..floor(n/5)} C(n-4*k, k) * 2^(n-2*k) *(-1)^k.
%F a(n) = A018922(n-3) for n >= 3. - _R. J. Mathar_, Mar 09 2007
%F First difference of A119407. - _Michael Somos_, Dec 28 2012
%F From _Petros Hadjicostas_, Jun 12 2019: (Start)
%F G.f.: 1/((1 - x)*(1 - x - x^2 - x^3 - x^4)).
%F 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).
%F (End)
%F a(n) = Sum_{k=0..n+3} A000078(k). - _Greg Dresden_, Jan 01 2021
%e 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 + ...
%t CoefficientList[Series[1/(1 - 2*z + z^5), {z, 0, 40}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2011 *)
%t LinearRecurrence[{2,0,0,0,-1}, {1,2,4,8,16}, 40] (* _G. C. Greubel_, Jun 12 2019 *)
%o (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 */
%o (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
%o (Sage) (1/(1-2*x+x^5)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 12 2019
%o (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
%Y Cf. A018922, A119407 (partial sums), A000078 (first differences).
%Y Cf. A209888. - _Alois P. Heinz_, Mar 14 2012
%Y Column k = 1 of array A140996 (with a different offset) and second main diagonal of A140995.
%Y Column k = 4 of A172119 (with a different offset).
%K easy,nonn
%O 0,2
%A _Paul Barry_, May 10 2005
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