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A106709
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Expansion of g.f. -2*x/(1 - 5*x + 2*x^2).
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3
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0, -2, -10, -46, -210, -958, -4370, -19934, -90930, -414782, -1892050, -8630686, -39369330, -179585278, -819187730, -3736768094, -17045465010, -77753788862, -354678014290, -1617882493726, -7380056440050, -33664517212798, -153562473183890, -700483331493854
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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
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MAPLE
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a:= n-> (<<0|-2>, <1|5>>^n)[1, 2]:
seq(a(n), n=0..25); # Alois P. Heinz, Nov 19 2020
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MATHEMATICA
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LinearRecurrence[{5, -2}, {0, -2}, 41] (* G. C. Greubel, Sep 10 2021 *)
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PROG
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(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
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CROSSREFS
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Cf. A107839, A337277.
Sequence in context: A080643 A032389 A290923 * A204091 A221196 A137193
Adjacent sequences: A106706 A106707 A106708 * A106710 A106711 A106712
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KEYWORD
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sign,easy,less
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AUTHOR
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Roger L. Bagula, May 30 2005
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EXTENSIONS
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Edited by N. J. A. Sloane, Apr 30 2006
New name by G. C. Greubel, Sep 10 2021
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STATUS
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approved
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