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A007179
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Dual pairs of integrals arising from reflection coefficients.
(Formerly M3284)
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8
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0, 1, 1, 4, 6, 16, 28, 64, 120, 256, 496, 1024, 2016, 4096, 8128, 16384, 32640, 65536, 130816, 262144, 523776, 1048576, 2096128, 4194304, 8386560, 16777216, 33550336, 67108864, 134209536, 268435456
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OFFSET
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0,4
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
J. Heading, Theorem relating to the development of a reflection coefficient in terms of a small parameter, J. Phys. A 14 (1981), 357-367.
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - N. J. A. Sloane, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).
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FORMULA
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From Paul Barry, Apr 28 2004: (Start)
Binomial transform is (A000244(n)+A001333(n))/2.
G.f.: x*(1-x)/((1-2*x)*(1-2*x^2)).
a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3).
a(n) = 2^n/2-2^(n/2)*(1+(-1)^n)/4. (End)
G.f.: (1+x*Q(0))*x/(1-x), where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
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MAPLE
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f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;
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MATHEMATICA
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LinearRecurrence[{2, 2, -4}, {0, 1, 1}, 30] (* Harvey P. Dale, Nov 30 2015 *)
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PROG
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(MAGMA) [Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // Vincenzo Librandi, Aug 20 2011
(PARI) Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ Michel Marcus, Jan 28 2016
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CROSSREFS
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Sequence in context: A059736 A261682 A102731 * A112576 A174804 A081487
Adjacent sequences: A007176 A007177 A007178 * A007180 A007181 A007182
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Simon Plouffe
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STATUS
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approved
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