A271566 a(n) is the length of the n-th run in A137734.

2, 3, 1, 1, 3, 1, 2, 4, 1, 3, 5, 1, 4, 6, 1, 5, 7, 1, 6, 8, 1, 7, 9, 1, 8, 10, 1, 9, 11, 1, 10, 12, 1, 11, 13, 1, 12, 14, 1, 13, 15, 1, 14, 16, 1, 15, 17, 1, 16, 18, 1, 17, 19, 1, 18, 20, 1, 19, 21, 1, 20, 22, 1, 21, 23, 1, 22, 24, 1, 23, 25

Offset

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2000 (first 945 terms from Alec Jones)

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Formula

For n > 2: a(n) = 1 if 3 divides n; a(n) = (1/3)*(n-1) if n == 1 (mod 3); a(n) = a(n-1) + 2 if n == 2 (mod 3).

For n > 2, a(n) = (4*n + 12 + (6-4*n)*cos(2*n*Pi/3) - 5*sqrt(3)*sin(2*n*Pi/3) + 5*sqrt(3)*sin(4*n*Pi/3))/18. - Wesley Ivan Hurt, Sep 25 2017

From Colin Barker, Sep 26 2017: (Start)

G.f.: x*(2 + 3*x + x^2 - 3*x^3 - 3*x^4 - x^5 + 2*x^6 + x^7) / ((1 - x)^2*(1 + x + x^2)^2).

a(n) = 2*a(n-3) - a(n-6) for n>8. (End)

Mathematica

Join[{2, 3}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 1, 3, 1, 2, 4}, 200]] (* Vincenzo Librandi, Sep 27 2017 *)

Prog

(PARI) Vec(x*(2 + 3*x + x^2 - 3*x^3 - 3*x^4 - x^5 + 2*x^6 + x^7) / ((1 - x)^2*(1 + x + x^2)^2) + O(x^100)) \\ Colin Barker, Sep 26 2017
(Magma) I:=[2, 3, 1, 1, 3, 1, 2, 4]; [n le 8 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..100]]; // Vincenzo Librandi, Sep 27 2017

Crossrefs

Cf. A137734.

Sequence in context: A300521 A356152 A317627 * A296659 A270823 A067627

Adjacent sequences: A271563 A271564 A271565 * A271567 A271568 A271569

Keyword

nonn,easy

Author

Alec Jones, Apr 12 2016

Status

approved