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A191010
a(n) = (n*4^(n+1) + (6*4^(n+1)+(-1)^n)/5)/5.
1
1, 7, 41, 215, 1065, 5079, 23593, 107479, 482345, 2139095, 9395241, 40936407, 177167401, 762356695, 3264175145, 13915694039, 59098749993, 250138895319, 1055531162665, 4442026976215, 18647717207081, 78109306037207, 326510972984361, 1362338887279575
OFFSET
0,2
COMMENTS
a(n) = 4^(n+1)*H(2^n)/5 with H(2^n) = n+(6+(-1)^n/4^(n+1))/5 = E(N(2^n)), where X, X(1), X(2),... denote random variables with pdf P(X = 1) = P(X = 4) = 1/5 and P(X = 2) = 3/5, N(x) is the first value of k such that X(1)*X(2)*...*X(k) > x and H(x)= E(N(x)).
FORMULA
a(n) = (n*4^(n+1) + (6*4^(n+1)+(-1)^n)/5)/5.
From Colin Barker, May 03 2017: (Start)
G.f.: 1 / ((1 + x)*(1 - 4*x)^2).
a(n) = 7*a(n-1) - 8*a(n-2) - 16*a(n-3) for n>2.
(End)
E.g.f.: (80*x*exp(4*x)+24*exp(4*x)+exp(-x))/25. - Robert Israel, May 03 2017
MAPLE
seq((n*4^(n+1) + (6*4^(n+1)+(-1)^n)/5)/5, n=0..50); # Robert Israel, May 03 2017
MATHEMATICA
CoefficientList[Series[1/((1 + x) (1 - 4 x)^2), {x, 0, 23}], x] (* or *)
LinearRecurrence[{7, -8, -16}, {1, 7, 41}, 24] (* Michael De Vlieger, May 03 2017 *)
PROG
(PARI) a(n)= (n*4^(n+1)+(6*4^(n+1)+(-1)^n)/5)/5; \\ Michel Marcus, Oct 16 2014
(PARI) Vec(1 / ((1 + x)*(1 - 4*x)^2) + O(x^30)) \\ Colin Barker, May 03 2017
CROSSREFS
Cf. A191008.
Sequence in context: A266887 A237664 A168584 * A239041 A081625 A144635
KEYWORD
nonn,easy
AUTHOR
Edward Omey, Jun 16 2011
EXTENSIONS
Formula corrected and more terms from Michel Marcus, Oct 16 2014
Edited by M. F. Hasler, Oct 16 2014
STATUS
approved