A140824 - OEIS

%I #29 Aug 08 2017 12:59:32

%S 0,1,3,6,15,41,108,281,735,1926,5043,13201,34560,90481,236883,620166,

%T 1623615,4250681,11128428,29134601,76275375,199691526,522799203,

%U 1368706081,3583319040,9381251041,24560434083,64300051206,168339719535,440719107401,1153817602668

%N Expansion of (x-x^3)/(1-3*x+2*x^2-3*x^3+x^4).

%C Case P1 = 3, P2 = 0, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 25 2014

%H G. C. Greubel, <a href="/A140824/b140824.txt">Table of n, a(n) for n = 0..1000</a>

%H Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a>, Integers, Volume 12A (2012) The John Selfridge Memorial Volume

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,3,-1).

%F a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 6, a(n) - 3 a(n + 1) + 2 a(n + 2) - 3 a(n + 3) + a(n + 4) = 0.

%F From _Peter Bala_, Mar 25 2014: (Start)

%F a(n) = 2/3*( T(n,3/2) - T(n,0) ), where T(n,x) is a Chebyshev polynomial of the first kind.

%F a(n) = 1/3 * (A005248(n) - (i^n + (-i)^n)) = 1/3 * (Fibonacci(2*n-1) + Fibonacci(2*n+1) - (i^n + (-i)^n)).

%F a(n) = bottom left entry of the 2 X 2 matrix 2*T(n, 1/2*M), where M is the 2 X 2 matrix [0, 0; 1, 3].

%F The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/sqrt(2)*(sqrt(5) + i)*x + x^2) and x/(1 - 1/sqrt(2)*(sqrt(5) - i)*x + x^2). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

%F a(n) = A099483(n) - A099483(n-2). - _R. J. Mathar_, Feb 10 2016

%t LinearRecurrence[{3, -2, 3, -1}, {0, 1, 3, 6}, 50] (* _G. C. Greubel_, Aug 08 2017 *)

%o (PARI) x='x+O('x^50); concat([0], Vec((x-x^3)/(1-3*x+2*x^2-3*x^3+x^4))) \\ _G. C. Greubel_, Aug 08 2017

%Y Cf. A006238, A005248, A054493, A078070, A092184, A098306, A100047, A100048, A108196, A138573, A152090, A218134.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Sep 07 2009, based on email from _R. K. Guy_, Mar 09 2009