A170931 Extended Lucas L(n,i) = n*(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n*(n+4)); a=(n+d)/2; b=(n-d)/2.

2, 4, 24, 112, 544, 2624, 12672, 61184, 295424, 1426432, 6887424, 33255424, 160571392, 775307264, 3743514624, 18075287552, 87275208704, 421401985024, 2034708774912, 9824443039744, 47436607258624, 229044201193472

Offset

0,1

Comments

Sequence gives the rational part of the radii of the circles 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(n)*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

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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.

Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man

Index entries for linear recurrences with constant coefficients, signature (4, 4).

Formula

From R. J. Mathar, Feb 05 2010: (Start)

a(n) = 2*A084128(n) = 4*a(n-1) + 4*a(n-2).

G.f.: 2*(1-2*x)/(1 - 4*x - 4*x^2). (End)

Example

L(n,0)=2, L(n,1)=n.

Mathematica

CoefficientList[Series[2 (1 - 2 x) / (1 - 4 x - 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 16 2015 *)
LinearRecurrence[{4, 4}, {2, 4}, 30] (* Harvey P. Dale, Sep 03 2016 *)

Prog

(PARI) x='x+O('x^30); Vec(2*(1-2*x)/(1 - 4*x - 4*x^2)) \\ G. C. Greubel, Dec 21 2017
(Magma) I:=[2, 4]; [n le 5 select I[n] else 4*Self(n-1)+4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017

Crossrefs

Cf. similar sequences with d=sqrt(n*(n+k)): A000032 (k=1, classic Lucas), A080040 (k=2), A085480 (k=3).

Cf. A094013, A174968.

Sequence in context: A119036 A192382 A232205 * A317999 A371892 A164313

Adjacent sequences: A170928 A170929 A170930 * A170932 A170933 A170934

Keyword

nonn,easy

Author

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Feb 04 2010

Status

approved