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 A094013 Expansion of (1-4x)/(1-4x-4x^2). 7
 1, 0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Inverse binomial transform of A000129(2n-1). a(n+2)/4=A057087(n). 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 LINKS Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5. Tanya Khovanova, Recursive Sequences Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man Index entries for linear recurrences with constant coefficients, signature (4,4). FORMULA a(n) = (2 + 2sqrt(2))^n(1/2 - sqrt(2)/4) + (2 - 2sqrt(2))^n(1/2 + sqrt(2)/4). a(n) = 4*a(n-1)+4*a(n-2); a(0)=1, a(1)=0. - Philippe Deléham, Nov 03 2008 a(n) = A057087(n)-4*A057087(n-1). - R. J. Mathar, Jan 15 2013 MATHEMATICA 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 *) PROG (PARI) Vec((1-4*x)/(1-4*x-4*x^2) + O(x^30)) \\ Michel Marcus, Feb 15 2015 CROSSREFS Cf. A001653, A170931, A174968. Sequence in context: A081682 A068788 * A106568 A183146 A160564 A075581 Adjacent sequences:  A094010 A094011 A094012 * A094014 A094015 A094016 KEYWORD easy,nonn AUTHOR Paul Barry, Apr 21 2004 STATUS approved

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