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%I #39 Sep 08 2022 08:45:13
%S 1,0,4,16,80,384,1856,8960,43264,208896,1008640,4870144,23515136,
%T 113541120,548225024,2647064576,12781158400,61712891904,297976201216,
%U 1438756372480,6946930294784,33542746669056,161958707855360
%N Expansion of (1-4*x)/(1-4*x-4*x^2).
%C Inverse binomial transform of A000129(2n-1). a(n+2)/4 = A057087(n).
%C a(n) is the irrational part of circle radii in nested circles and squares inspired by Vitruvian Man, starting with a square whose sides are of length 4 (in some units). The radius of the circle is an integer in the real quadratic number field Q(sqrt(2)), namely R(n) = A(n-1) + B(m)*sqrt(2) with A(-1)=1, for n >= 1, A(n-1) = A170931(n-1)*-1^(n-1); and B(n) = A094013(n)*-1^n. See illustrations in the links. - _Kival Ngaokrajang_, Feb 15 2015
%H G. C. Greubel, <a href="/A094013/b094013.txt">Table of n, a(n) for n = 0..1000</a>
%H Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.pdf">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Kival Ngaokrajang, <a href="/A094013/a094013.pdf">Illustration of initial terms</a>, <a href="/A094013/a094013_1.pdf">Vitruvian Man</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,4).
%F a(n) = (2 + 2*sqrt(2))^n*(1/2 - sqrt(2)/4) + (2 - 2*sqrt(2))^n*(1/2 + sqrt(2)/4).
%F a(n) = 4*a(n-1) + 4*a(n-2); a(0)=1, a(1)=0. - _Philippe Deléham_, Nov 03 2008
%F a(n) = A057087(n) - 4*A057087(n-1). - _R. J. Mathar_, Jan 15 2013
%F From _G. C. Greubel_, Dec 04 2021: (Start)
%F a(n) = 2^n * A000129(n-1).
%F E.g.f.: exp(2*x)*( cosh(2*sqrt(2)*x) - (1/sqrt(2))*sinh(2*sqrt(2)*x) ). (End)
%t CoefficientList[Series[(1-4x)/(1-4x-4x^2),{x,0,40}],x] (* or *) LinearRecurrence[{4,4},{1,0},40] (* _Harvey P. Dale_, May 21 2012 *)
%t Table[2^n*Fibonacci[n-1, 2], {n, 0, 40}] (* _G. C. Greubel_, Dec 04 2021 *)
%o (PARI) Vec((1-4*x)/(1-4*x-4*x^2) + O(x^30)) \\ _Michel Marcus_, Feb 15 2015
%o (Magma) [n le 2 select 2-n else 4*(Self(n-1) + Self(n-2)): n in [1..41]]; // _G. C. Greubel_, Dec 04 2021
%o (Sage) [2^n*lucas_number1(n-1, 2, -1) for n in (0..40)] # _G. C. Greubel_, Dec 04 2021
%Y Cf. A000129, A001653, A170931, A174968.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Apr 21 2004
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