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 A106709 Expansion of g.f. -2*x/(1 - 5*x + 2*x^2). 3
 0, -2, -10, -46, -210, -958, -4370, -19934, -90930, -414782, -1892050, -8630686, -39369330, -179585278, -819187730, -3736768094, -17045465010, -77753788862, -354678014290, -1617882493726, -7380056440050, -33664517212798, -153562473183890, -700483331493854 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Let T(n,k) denote the k-th element of row n of Stern's triangle (see A337277). Then b(n) = Sum_k T(n,k)*T(n,k+1) gives the present sequence (without the signs). - N. J. A. Sloane, Nov 19 2020 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. Also American Mathematical Monthly 127.2 (2020): 99-111. Index entries for linear recurrences with constant coefficients, signature (5,-2). FORMULA a(n) = -2*A107839(n-1), n>0. a(n) = first entry of v(n), where v(n) = M*v(n-1), M is the 2 X 2 matrix ({0, -2}, {1, 5}) and v(0) is the column vector (0, 1). G.f.: -2*x/(1-5*x+2*x^2). - Alois P. Heinz, Nov 26 2020 a(n) = -sqrt(2)^(n+1)*ChebyshevU(n-1, 5/(2*sqrt(2))). - G. C. Greubel, Sep 10 2021 MAPLE a:= n-> (<<0|-2>, <1|5>>^n)[1, 2]: seq(a(n), n=0..25);  # Alois P. Heinz, Nov 19 2020 MATHEMATICA LinearRecurrence[{5, -2}, {0, -2}, 41] (* G. C. Greubel, Sep 10 2021 *) PROG (MAGMA) I:=[0, -2]; [n le 2 select I[n] else 5*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Sep 10 2021 (Sage) [-round(sqrt(2)^(n+1)*chebyshev_U(n-1, 5/(2*sqrt(2)))) for n in (0..40)] # G. C. Greubel, Sep 10 2021 CROSSREFS Cf. A107839, A337277. Sequence in context: A080643 A032389 A290923 * A204091 A221196 A137193 Adjacent sequences:  A106706 A106707 A106708 * A106710 A106711 A106712 KEYWORD sign,easy,less AUTHOR Roger L. Bagula, May 30 2005 EXTENSIONS Edited by N. J. A. Sloane, Apr 30 2006 New name by G. C. Greubel, Sep 10 2021 STATUS approved

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Last modified July 3 03:28 EDT 2022. Contains 355030 sequences. (Running on oeis4.)