A094013 Expansion of (1-4*x)/(1-4*x-4*x^2).

1, 0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360

Offset: 0

Author

Paul Barry, Apr 21 2004

Keywords

• easy
• nonn

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Martin Burtscher, Igor Szczyrba, and 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).

Crossrefs

Cf. A000129, A001653, A170931, A174968.

Programs

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

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 *)
Table[2^n*Fibonacci[n-1, 2], {n, 0, 40}] (* G. C. Greubel, Dec 04 2021 *)

PARI

Vec((1-4*x)/(1-4*x-4*x^2) + O(x^30)) \\ Michel Marcus, Feb 15 2015

Sage

[2^n*lucas_number1(n-1, 2, -1) for n in (0..40)] # G. C. Greubel, Dec 04 2021

Formula

a(n) = (2 + 2*sqrt(2))^n*(1/2 - sqrt(2)/4) + (2 - 2*sqrt(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

From G. C. Greubel, Dec 04 2021: (Start)

a(n) = 2^n * A000129(n-1).

E.g.f.: exp(2*x)*( cosh(2*sqrt(2)*x) - (1/sqrt(2))*sinh(2*sqrt(2)*x) ). (End)