A106709 Expansion of g.f. -2*x/(1 - 5*x + 2*x^2).

0, -2, -10, -46, -210, -958, -4370, -19934, -90930, -414782, -1892050, -8630686, -39369330, -179585278, -819187730, -3736768094, -17045465010, -77753788862, -354678014290, -1617882493726, -7380056440050, -33664517212798, -153562473183890, -700483331493854

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