This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007179 M3284 #49 Jun 25 2023 21:07:20
%S A007179 0,1,1,4,6,16,28,64,120,256,496,1024,2016,4096,8128,16384,32640,65536,
%T A007179 130816,262144,523776,1048576,2096128,4194304,8386560,16777216,
%U A007179 33550336,67108864,134209536,268435456,536854528,1073741824,2147450880,4294967296,8589869056
%N A007179 Dual pairs of integrals arising from reflection coefficients.
%D A007179 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007179 Vincenzo Librandi, <a href="/A007179/b007179.txt">Table of n, a(n) for n = 0..200</a>
%H A007179 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A007179 J. Heading, <a href="http://dx.doi.org/10.1088/0305-4470/14/2/011">Theorem relating to the development of a reflection coefficient in terms of a small parameter</a>, J. Phys. A 14 (1981), 357-367.
%H A007179 Kyu-Hwan Lee, Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%H A007179 A. Yajima, <a href="https://www.jstage.jst.go.jp/article/bcsj/87/11/87_20140204/_pdf">How to calculate the number of stereoisomers of inositol-homologs</a>, 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
%H A007179 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-4).
%F A007179 From _Paul Barry_, Apr 28 2004: (Start)
%F A007179 Binomial transform is (A000244(n)+A001333(n))/2.
%F A007179 G.f.: x*(1-x)/((1-2*x)*(1-2*x^2)).
%F A007179 a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3).
%F A007179 a(n) = 2^n/2-2^(n/2)*(1+(-1)^n)/4. (End)
%F A007179 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
%F A007179 a(n) = A011782(n+2) - A077957(n) - _Gus Wiseman_, Feb 26 2022
%e A007179 From _Gus Wiseman_, Feb 26 2022: (Start)
%e A007179 Also the number of integer compositions of n with at least one odd part. For example, the a(1) = 1 through a(5) = 16 compositions are:
%e A007179 (1) (1,1) (3) (1,3) (5)
%e A007179 (1,2) (3,1) (1,4)
%e A007179 (2,1) (1,1,2) (2,3)
%e A007179 (1,1,1) (1,2,1) (3,2)
%e A007179 (2,1,1) (4,1)
%e A007179 (1,1,1,1) (1,1,3)
%e A007179 (1,2,2)
%e A007179 (1,3,1)
%e A007179 (2,1,2)
%e A007179 (2,2,1)
%e A007179 (3,1,1)
%e A007179 (1,1,1,2)
%e A007179 (1,1,2,1)
%e A007179 (1,2,1,1)
%e A007179 (2,1,1,1)
%e A007179 (1,1,1,1,1)
%e A007179 (End)
%p A007179 f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;
%t A007179 LinearRecurrence[{2,2,-4},{0,1,1},30] (* _Harvey P. Dale_, Nov 30 2015 *)
%t A007179 Table[2^(n-1)-If[EvenQ[n],2^(n/2-1),0],{n,0,15}] (* _Gus Wiseman_, Feb 26 2022 *)
%o A007179 (Magma) [Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // _Vincenzo Librandi_, Aug 20 2011
%o A007179 (PARI) Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ _Michel Marcus_, Jan 28 2016
%Y A007179 Column k=2 of A309748.
%Y A007179 Odd bisection is A000302.
%Y A007179 Even bisection is A006516 = 2^(n-1)*(2^n - 1).
%Y A007179 The complement is counted by A077957, internal version A027383.
%Y A007179 The internal case is A274230, even bisection A134057.
%Y A007179 A000045(n-1) counts compositions without odd parts, non-singleton A077896.
%Y A007179 A003242 counts Carlitz compositions.
%Y A007179 A011782 counts compositions.
%Y A007179 A034871, A097805, and A345197 count compositions by alternating sum.
%Y A007179 A052952 (or A074331) counts non-singleton compositions without even parts.
%Y A007179 Cf. A000918, A033484, A052955, A060867, A116406, A138364.
%K A007179 nonn,easy
%O A007179 0,4
%A A007179 _N. J. A. Sloane_, _Simon Plouffe_