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 A106435 a(n) = 3*a(n-1) + 3*a(n-2), a(0)=0, a(1)=3. 12
 0, 3, 9, 36, 135, 513, 1944, 7371, 27945, 105948, 401679, 1522881, 5773680, 21889683, 82990089, 314639316, 1192888215, 4522582593, 17146412424, 65006985051, 246460192425, 934401532428, 3542585174559, 13430960120961 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The first entry of the vector v[n] = M*v[n-1], where M is the 2 x 2 matrix [[0,3],[1,3]] and v[1] is the column vector [0,1]. The characteristic polynomial of the matrix M is x^2-3x-3. LINKS Reinhard Zumkeller, 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 Index entries for linear recurrences with constant coefficients, signature (3,3). FORMULA G.f.: 3*x/(1-3*x-3*x^2). - Philippe Deléham, Nov 19 2008 From G. C. Greubel, Mar 12 2020: (Start) a(n) = 3^((n+1)/2) * Fibonacci(n, sqrt(3)), where F(n, x) is the Fibonacci polynomial. a(n) = 3^((n+1)/2)*i^(1-n)*ChebyshevU(n-1, i*sqrt(3)/2). (End) MAPLE seq(coeff(series(3*x/(1-3*x-3*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Mar 12 2020 MATHEMATICA LinearRecurrence[{3, 3}, {0, 3}, 30] (* G. C. Greubel, Mar 12 2020 *) PROG (PARI) a(n)=([0, 3; 1, 3]^n)[1, 2] (Haskell) a106435 n = a106435_list !! n a106435_list = 0 : 3 : map (* 3) (zipWith (+) a106435_list (tail a106435_list)) -- Reinhard Zumkeller, Oct 15 2011 (Magma) a:=[0, 3]; [n le 2 select a[n] else 3*Self(n-1) + 3*Self(n-2) : n in [1..24]]; // Marius A. Burtea, Jan 21 2020 (Magma) R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!(3*x/(1-3*x-3*x^2))); // Marius A. Burtea, Jan 21 2020 (Sage) [3^((n+1)/2)*i^(1-n)*chebyshev_U(n-1, i*sqrt(3)/2) for n in (0..30)] # G. C. Greubel, Mar 12 2020 CROSSREFS Equals 3*A030195(n). Cf. A028860. Cf. A002605, A026150, A028859, A080040, A083337, A108898, A125145. Sequence in context: A057390 A183495 A185162 * A276368 A058540 A350451 Adjacent sequences: A106432 A106433 A106434 * A106436 A106437 A106438 KEYWORD nonn,easy AUTHOR Roger L. Bagula, May 29 2005 EXTENSIONS Edited by N. J. A. Sloane, May 20 2006 and May 29 2006 Offset corrected by Reinhard Zumkeller, Oct 15 2011 STATUS approved

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Last modified September 10 21:37 EDT 2024. Contains 375795 sequences. (Running on oeis4.)