OFFSET
0,2
COMMENTS
Solutions (x, y) = (a(n), a(n+1)) satisfying x^2 + y^2 = 3xy - 4. - Michel Lagneau, Feb 01 2014
Except for the first term, positive values of x (or y) satisfying x^2 - 18xy + y^2 + 256 = 0. - Colin Barker, Feb 16 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Youngwoo Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, arXiv:1804.08119 [math.NT], 2018.
D. Yaqubi, M. Farrokhi D.G., H. Gahsemian Zoeram, Lattice paths inside a table. I, arXiv:1612.08697 [math.CO], 2016-2017.
Index entries for linear recurrences with constant coefficients, signature (3,-1).
FORMULA
From Colin Barker, Feb 01 2014: (Start)
a(n) = 3*a(n-1) - a(n-2) for n > 0.
G.f.: (1 -x -x^2)/(1-3*x+x^2). (End)
a(n) = 2*A001519(n) for n > 0. - Colin Barker, Feb 04 2014
From G. C. Greubel, Jan 22 2020: (Start)
a(n) = 2*(ChebyshevU(n, 3/2) - 2*ChebyshevU(n-1, 3/2)), with a(0)=1.
E.g.f.: -1 + 2*exp(3*x/2)*( cosh(sqrt(5)*x/2) - sinh(sqrt(5)*x/2)/sqrt(5) ). (End)
MAPLE
seq(`if`(n=0, 1, simplify(2*(ChebyshevU(n, 3/2)-2*ChebyshevU(n-1, 3/2)))), n = 0..30); # G. C. Greubel, Jan 22 2020
MATHEMATICA
CoefficientList[Series[(1-x-x^2)/(1-3*x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 05 2014 *)
Join[{1}, LinearRecurrence[{3, -1}, {2, 4}, 30]] (* Harvey P. Dale, Oct 01 2014 *)
Table[If[n==0, 1, 2*(ChebyshevU[n, 3/2] -2*ChebyshevU[n-1, 3/2])], {n, 0, 30}] (* G. C. Greubel, Jan 22 2020 *)
PROG
(PARI) Vec((1-x-x^2)/(1-3*x+x^2) + O(x^40)) \\ Colin Barker, Feb 01 2014
(Magma) I:=[2, 4]; [1] cat [n le 2 select I[n] else 3*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 22 2020
(Sage) [1]+[2*(chebyshev_U(n, 3/2) -2*chebyshev_U(n-1, 3/2)) for n in (1..30)] # G. C. Greubel, Jan 22 2020
(GAP) a:=[2, 4];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Jan 22 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 28 2000
STATUS
approved