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

Author

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

Keywords

• nonn
• easy

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

Examples

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

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).

Crossrefs

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

Cf. A094013, A174968.

Programs

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

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 *)

PARI

x='x+O('x^30); Vec(2*(1-2*x)/(1 - 4*x - 4*x^2)) \\ G. C. Greubel, Dec 21 2017

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)