OFFSET
0,3
COMMENTS
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).
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
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)
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 *)
PROG
(PARI) Vec((1-4*x)/(1-4*x-4*x^2) + O(x^30)) \\ Michel Marcus, Feb 15 2015
(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
(Sage) [2^n*lucas_number1(n-1, 2, -1) for n in (0..40)] # G. C. Greubel, Dec 04 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 21 2004
STATUS
approved