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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

Table of n, a(n) for n=0..22.

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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Last modified May 28 00:50 EDT 2017. Contains 287210 sequences.