A055819 Row sums of array T in A055818; twice the odd-indexed Fibonacci numbers after initial term.

1, 2, 4, 10, 26, 68, 178, 466, 1220, 3194, 8362, 21892, 57314, 150050, 392836, 1028458, 2692538, 7049156, 18454930, 48315634, 126491972, 331160282, 866988874, 2269806340, 5942430146, 15557484098, 40730022148, 106632582346

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

Essentially the same as A052995.

Sequence in context: A149810 A095337 A162533 * A052995 A113337 A084575

Adjacent sequences: A055816 A055817 A055818 * A055820 A055821 A055822

Keyword

nonn,easy

Author

Clark Kimberling, May 28 2000

Status

approved