This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A170931 #38 Sep 08 2022 08:45:49
%S A170931 2,4,24,112,544,2624,12672,61184,295424,1426432,6887424,33255424,
%T A170931 160571392,775307264,3743514624,18075287552,87275208704,421401985024,
%U A170931 2034708774912,9824443039744,47436607258624,229044201193472
%N 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.
%C A170931 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
%H A170931 Harvey P. Dale, <a href="/A170931/b170931.txt">Table of n, a(n) for n = 0..1000</a>
%H A170931 Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A170931 Kival Ngaokrajang, <a href="/A170931/a170931.pdf">Illustration of initial terms</a>, <a href="/A170931/a170931_1.pdf">Vitruvian Man</a>
%H A170931 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, 4).
%F A170931 From _R. J. Mathar_, Feb 05 2010: (Start)
%F A170931 a(n) = 2*A084128(n) = 4*a(n-1) + 4*a(n-2).
%F A170931 G.f.: 2*(1-2*x)/(1 - 4*x - 4*x^2). (End)
%e A170931 L(n,0)=2, L(n,1)=n.
%t A170931 CoefficientList[Series[2 (1 - 2 x) / (1 - 4 x - 4 x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 16 2015 *)
%t A170931 LinearRecurrence[{4,4},{2,4},30] (* _Harvey P. Dale_, Sep 03 2016 *)
%o A170931 (PARI) x='x+O('x^30); Vec(2*(1-2*x)/(1 - 4*x - 4*x^2)) \\ _G. C. Greubel_, Dec 21 2017
%o A170931 (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
%Y A170931 Cf. similar sequences with d=sqrt(n*(n+k)): A000032 (k=1, classic Lucas), A080040 (k=2), A085480 (k=3).
%Y A170931 Cf. A094013, A174968.
%K A170931 nonn,easy
%O A170931 0,1
%A A170931 Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Feb 04 2010