A128205 a(n) = 2^(n-1)*A047240(n).

0, 1, 4, 24, 56, 128, 384, 832, 1792, 4608, 9728, 20480, 49152, 102400, 212992, 491520, 1015808, 2097152, 4718592, 9699328, 19922944, 44040192, 90177536, 184549376, 402653184, 822083584, 1677721600, 3623878656, 7381975040, 15032385536, 32212254720

Offset

0,3

Comments

-a(n) is the Hankel transform of A030662(n) = binomial(2*n,n)-1.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,8,-16).

Formula

a(n) = 2^(n-1)*(cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3)/3 + 2n - 1).

From R. J. Mathar, Apr 28 2008: (Start)

O.g.f.: x*(1+2*x+16*x^2)/((2*x-1)^2*(4*x^2+2*x+1)).

a(n) = 2*a(n-1) + 8*a(n-3) - 16*a(n-4). (End)

Sum_{n>=1} 1/a(n) = arctan(sqrt(3)/5)/(2*sqrt(3)) + arctan(sqrt(6))/sqrt(6) + arctanh(9*sqrt(2)/13)/(3*sqrt(2)) + log(2) - log(7)/4. - Amiram Eldar, Jan 22 2026

Mathematica

a047240[n_] := 6 Floor[n/3] + Mod[n, 3]
a128205[n_] := Map[2^(#-1) a047240[#]&, Range[0, n]]
a128205[25] (* data *) (* Hartmut F. W. Hoft, Mar 13 2017 *)
LinearRecurrence[{2, 0, 8, -16}, {0, 1, 4, 24}, 40] (* Harvey P. Dale, Feb 13 2024 *)

Prog

(PARI) concat(0, Vec(x*(1 + 2*x + 16*x^2) / ((1 - 2*x)^2*(1 + 2*x + 4*x^2)) + O(x^40))) \\ Colin Barker, Mar 13 2017

Crossrefs

Cf. A047240, A030662.

Sequence in context: A353250 A191778 A157625 * A085250 A166870 A124350

Adjacent sequences: A128202 A128203 A128204 * A128206 A128207 A128208

Keyword

easy,nonn

Author

Paul Barry, Feb 19 2007

Status

approved