A191007 a(n) = n*2^(n+1) + (2^(n+3)+(-1)^n)/3.

3, 9, 27, 69, 171, 405, 939, 2133, 4779, 10581, 23211, 50517, 109227, 234837, 502443, 1070421, 2271915, 4805973, 10136235, 21321045, 44739243, 93672789, 195734187, 408245589, 850045611, 1767200085, 3668617899, 7605671253, 15748213419, 32570168661, 67287820971

Offset: 0

Author

Edward Omey, Jun 16 2011

Keywords

• nonn
• easy
• less

Comments

Another renewal type of sequence: Let X, X(1),X(2),... denote independent random variables with pdf P(X=1) = P(X=2) = P(X=4) = 1/3. Let N(x) denote the first value of k such that X(1)*X(2)...*X(k) > x, and let H(x) = E(N(x)). The sequence a(n) is given by a(n) = 2^(n+1)*H(2^n).

Links

• Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Crossrefs

3 times A045883.

Programs

Magma

[n*2^(n+1)+(2^(n+3)+(-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

Mathematica

Table[n 2^(n + 1) + (2^(n + 3) + (-1)^n)/3, {n, 0, 70}] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{3,0,-4},{3,9,27},40] (* Harvey P. Dale, Feb 11 2024 *)

PARI

a(n) = n*2^(n+1) + (2^(n+3)+(-1)^n)/3; \\ Michel Marcus, Oct 16 2014

Formula

a(n) = n*2^(n+1) + (2^(n+3)+(-1)^n)/3.

a(n) = 3 * A045883(n+1).

G.f.: 3/((1 + x)*(1 - 2*x)^2). [Bruno Berselli, Oct 16 2014]

Extensions

Formula corrected and more terms from Michel Marcus, Oct 16 2014