OFFSET
0,4
COMMENTS
Limiting ratio after n=30 terms is 2.369205407038926.
With a(0)=0, this is a divisibility sequence with g.f. x(1-x^2)/(1 - x - 3x^2 - x^3 + x^4). The limiting ratio is the largest zero of 1 - x - 3x^2 - x^3 + x^4. - T. D. Noe, Dec 22 2008
The sequence is the case P1 = 1, P2 = -5, 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.
Index entries for linear recurrences with constant coefficients, signature (1,3,1,-1).
FORMULA
From Colin Barker, Jan 05 2014: (Start)
a(n) = a(n-1) +3*a(n-2) +a(n-3) -a(n-4) for n>4.
G.f.: (x^4-2*x^3-3*x^2+1) / (x^4-x^3-3*x^2-x+1). (End)
From Peter Bala, Mar 25 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = 1/4*(1 + sqrt(21)), beta = 1/4*(1 - sqrt(21)) and where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = U(n-1,1/4*(1 + sqrt(-3)))*U(n-1,1/4*(1 - sqrt(-3))) for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/4; 1, 1/2]. See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
MATHEMATICA
bb = Table[FullSimplify[ExpandAll[Product[1 + 4*Cos[k*Pi/n]^2 + 16*Cos[k*Pi/n]^4, {k, 1, (n - 1)/2}]]], {n, 0, 30}]
LinearRecurrence[{1, 3, 1, -1}, {1, 1, 1, 3, 7}, 50] (* G. C. Greubel, Aug 08 2017 *)
PROG
(PARI) Vec((x^4-2*x^3-3*x^2+1)/(x^4-x^3-3*x^2-x+1) + O(x^100)) \\ Colin Barker, Jan 05 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula and Gary W. Adamson, Nov 23 2008
STATUS
approved