A191008 a(n) = (n*3^(n+1)+((5*3^(n+1)+(-1)^(n))/4))/4.

1, 5, 22, 86, 319, 1139, 3964, 13532, 45517, 151313, 498226, 1627538, 5281195, 17039327, 54705208, 174877304, 556916953, 1767605981, 5593383310, 17651846030, 55570626391, 174557144075, 547207226932, 1712229064916, 5348509347109, 16680994498409, 51949382866474

Offset: 0

Author

Edward Omey, Jun 16 2011

Keywords

• nonn
• easy

Comments

Another renewal type of sequence. Let X, X(1), X(2),... denote random variables with pdf P(X = 1) = P(X = 4 ) = 1/4 and P(X = 2) = 1/2. 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 is given by a(n) = 3^(n+1)*H(2^n)/4.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-3,-9).

Programs

Maple

A191008:=n->(n*3^(n+1)+((5*3^(n+1)+(-1)^(n))/4))/4: seq(A191008(n), n=0..40); # Wesley Ivan Hurt, May 03 2017

Mathematica

LinearRecurrence[{5, -3, -9}, {1, 5, 22}, 27] (* or *)
CoefficientList[Series[1/((1 + x) (1 - 3 x)^2), {x, 0, 26}], x] (* Michael De Vlieger, May 03 2017 *)

PARI

a(n)=(n*3^(n+1)+((5*3^(n+1)+(-1)^(n))/4))/4; \\ Michel Marcus, Oct 16 2014

PARI

Vec(1 / ((1 + x)*(1 - 3*x)^2) + O(x^30)) \\ Colin Barker, May 03 2017

Formula

a(n) = (n*3^(n+1)+((5*3^(n+1)+(-1)^(n))/4))/4.

From Colin Barker, May 03 2017: (Start)

G.f.: 1 / ((1 + x)*(1 - 3*x)^2).

a(n) = 5*a(n-1) - 3*a(n-2) - 9*a(n-3) for n>2.

(End)

Extensions

More terms from Michel Marcus, Oct 16 2014