OFFSET
1,2
COMMENTS
a(n)*(-1)^(n+1) is the r=-3 member of the r-family of sequences S_r(n), n>=1, defined in A092184 where more information can be found.
The sequence is the case P1 = 3, P2 = -10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
J. C. G. Nottrot, Vierkantenkransen rond een driehoek, Pythagoras (Netherlands), 14 (1975-1976) 77-81.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
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 (4,4,-1).
FORMULA
G.f.: x*(1-x)/((1+x)*(1-5*x+x^2)).
a(n) = 4*a(n-1) + 4*a(n-2) - a(n-3), a(1)=1, a(2)=3, a(3)=16.
a(n) = (2/7)*(T(n, 5/2) - (-1)^n) with twice Chebyshev's polynomials of the first kind evaluated at x=5/2: 2*T(n, 5/2) = A003501(n) = ((5+sqrt(21))^n + (5-sqrt(21))^n)/2^n. - Wolfdieter Lang, Oct 18 2004
From Peter Bala, Apr 03 2014: (Start)
a(n) = |U(n-1, sqrt(3)*i/2)|^2, 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, 5/2; 1, 3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
MAPLE
A005386:=-(-1+z)/(z+1)/(z**2-5*z+1); [Conjectured by Simon Plouffe in his 1992 dissertation.]
a:= n-> (Matrix([[0, 1, 3]]). Matrix(3, (i, j)-> if (i=j-1) then 1 elif j=1 then [4, 4, -1][i] else 0 fi)^(n))[1, 1]: seq(a(n), n=1..25); # Alois P. Heinz, Aug 05 2008
MATHEMATICA
a[n_]:= Module[{n1=1, n2=0}, Do[{n1, n2}={Sqrt[3]*n1+n2, n1}, {n-1}]; n1^2];
Table[a[n], {n, 30}]
a[n_]:= Round[((5+Sqrt[21])/2)^n/7]; Table[a[n], {n, 30}]
Rest@(CoefficientList[Series[x/(1-x*(Sqrt[3]+x)), {x, 0, 30}], x])^2
Abs[ChebyshevU[Range[1, 40]-1, I*Sqrt[3]/2]]^2 (* G. C. Greubel, Nov 16 2022 *)
PROG
(Magma) I:=[1, 3, 16]; [n le 3 select I[n] else 4*Self(n-1) +4*Self(n-2) -Self(n-3): n in [1..41]]; // G. C. Greubel, Nov 16 2022
(SageMath)
def A005386(n): return abs(chebyshev_U(n-1, i*sqrt(3)/2))^2
[A005386(n) for n in range(1, 40)] # G. C. Greubel, Nov 16 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jean Meeus
EXTENSIONS
Edited by Peter J. C. Moses, Apr 23 2004
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
STATUS
approved