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 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. 5
 2, 4, 24, 112, 544, 2624, 12672, 61184, 295424, 1426432, 6887424, 33255424, 160571392, 775307264, 3743514624, 18075287552, 87275208704, 421401985024, 2034708774912, 9824443039744, 47436607258624, 229044201193472 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 11 19:17 EDT 2024. Contains 375073 sequences. (Running on oeis4.)