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A140824
Expansion of (x-x^3)/(1-3*x+2*x^2-3*x^3+x^4).
1
0, 1, 3, 6, 15, 41, 108, 281, 735, 1926, 5043, 13201, 34560, 90481, 236883, 620166, 1623615, 4250681, 11128428, 29134601, 76275375, 199691526, 522799203, 1368706081, 3583319040, 9381251041, 24560434083, 64300051206, 168339719535, 440719107401, 1153817602668
OFFSET
0,3
COMMENTS
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
LINKS
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume
FORMULA
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.
From Peter Bala, Mar 25 2014: (Start)
a(n) = 2/3*( T(n,3/2) - T(n,0) ), where T(n,x) is a Chebyshev polynomial of the first kind.
a(n) = 1/3 * (A005248(n) - (i^n + (-i)^n)) = 1/3 * (Fibonacci(2*n-1) + Fibonacci(2*n+1) - (i^n + (-i)^n)).
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].
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)
a(n) = A099483(n) - A099483(n-2). - R. J. Mathar, Feb 10 2016
MATHEMATICA
LinearRecurrence[{3, -2, 3, -1}, {0, 1, 3, 6}, 50] (* G. C. Greubel, Aug 08 2017 *)
PROG
(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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Sep 07 2009, based on email from R. K. Guy, Mar 09 2009
STATUS
approved