A168052 Hankel transform of a Motzkin variant.

1, -1, 2, -3, 3, -4, 5, -5, 6, -7, 7, -8, 9, -9, 10, -11, 11, -12, 13, -13, 14, -15, 15, -16, 17, -17, 18, -19, 19, -20, 21, -21, 22, -23, 23, -24, 25, -25, 26, -27, 27, -28, 29, -29, 30, -31, 31, -32, 33, -33, 34, -35, 35, -36, 37, -37, 38, -39, 39, -40, 41, -41, 42, -43

Offset

0,3

Comments

Hankel transform of A168051.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-1,0,-1,-1).

Formula

G.f.: (1+x^2)/((1+x)^2*(1-x+x^2)).

a(n) = cos(Pi*n/3)/3 + sqrt(3)*sin(Pi*n/3)/9 + 2*(n+1)*(-1)^n/3.

a(n) = A010892(n)/3 + 2*(-1)^n*(n+1)/3. - R. J. Mathar, Sep 30 2012

E.g.f.: exp(-x)*(6 - 6*x + exp(3*x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Apr 03 2023

Mathematica

LinearRecurrence[{-1, 0, -1, -1}, {1, -1, 2, -3}, 100] (* G. C. Greubel, Jul 07 2016 *)
CoefficientList[Series[(1 + x^2) / ((1 + x)^2 (1 - x + x^2)), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 08 2016 *)

Prog

(Magma) I:=[1, -1, 2, -3]; [n le 4 select I[n] else - Self(n-1)-Self(n-3)- Self(n-4): n in [1..65]]; // Vincenzo Librandi, Jul 08 2016

Crossrefs

Cf. A010892, A168051.

Sequence in context: A093878 A317686 A156689 * A131737 A004396 A066481

Adjacent sequences: A168049 A168050 A168051 * A168053 A168054 A168055

Keyword

easy,sign

Author

Paul Barry, Nov 17 2009

Status

approved