A100525 Bisection of A048654.

4, 22, 128, 746, 4348, 25342, 147704, 860882, 5017588, 29244646, 170450288, 993457082, 5790292204, 33748296142, 196699484648, 1146448611746, 6681992185828, 38945504503222, 226991034833504, 1323000704497802, 7711013192153308, 44943078448422046

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.de), Nov 24 2004

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (6,-1).

Crossrefs

Cf. A001109, A038761, A048654.

Programs

Magma

I:=[4,22,128]; [n le 3 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 13 2015

Mathematica

CoefficientList[Series[(4-2x)/(1-6x+x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 13 2015 *)
LinearRecurrence[{6,-1},{4,22},30] (* Harvey P. Dale, Mar 25 2016 *)

PARI

Vec((4-2*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 13 2015

SageMath

[2*(2*chebyshev_U(n,3) - chebyshev_U(n-1,3)) for n in (0..30)] # G. C. Greubel, Jun 28 2022

Formula

G.f.: 2*(2-x)/(1-6*x+x^2). - Philippe Deléham, Nov 17 2008

a(0)=4, a(1)=22, a(n) = 6*a(n-1) - a(n-2) for n>1. - Philippe Deléham, Sep 19 2009

a(n) = 2*A038725(n+1). - R. J. Mathar, Sep 27 2014

a(n) = ( (5 + 4*sqrt(2))*(3 + 2*sqrt(2))^n - (5 - 4*sqrt(2))*(3 - 2*sqrt(2))^n )/(2*sqrt(2)). - Colin Barker, Oct 13 2015

From G. C. Greubel, Jun 28 2022: (Start)

a(n) = 2*( 2*ChebyshevU(n, 3) - ChenyshevU(n-1, 3) ).

E.g.f.: 2*exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (5/(2*sqrt(2)))*sinh(2*sqrt(2)*x) ). (End)