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A007179 Dual pairs of integrals arising from reflection coefficients.
(Formerly M3284)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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).

FORMULA

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

MAPLE

f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;

MATHEMATICA

LinearRecurrence[{2, 2, -4}, {0, 1, 1}, 30] (* Harvey P. Dale, Nov 30 2015 *)

PROG

(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

CROSSREFS

Sequence in context: A059736 A261682 A102731 * A112576 A174804 A081487

Adjacent sequences:  A007176 A007177 A007178 * A007180 A007181 A007182

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified February 16 08:26 EST 2019. Contains 320159 sequences. (Running on oeis4.)