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

Author

Clark Kimberling, May 28 2000

Keywords

• nonn
• easy

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).

Crossrefs

Essentially the same as A052995.

Programs

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

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

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 *)

PARI

Vec((1-x-x^2)/(1-3*x+x^2) + O(x^40)) \\ Colin Barker, Feb 01 2014

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

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)