OFFSET
1,4
COMMENTS
a(n-1) is the number of compositions of n with at least one part >= 4. - Joerg Arndt, Aug 06 2012
REFERENCES
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..300
David Broadhurst, Multiple Landen values and the tribonacci numbers, arXiv:1504.05303 [hep-th], 2015.
Simon Cowell, A Formula for the Reliability of a d-dimensional Consecutive-k-out-of-n:F System, arXiv preprint arXiv:1506.03580 [math.CO], 2015.
Erich Friedman, Illustration of initial terms
T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
Eric Weisstein's World of Mathematics, Run
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-2).
FORMULA
a(n) = 2^n - tribonacci(n+3), see A000073. - Vladeta Jovovic, Feb 23 2003
G.f.: x^3/((1-2*x)*(1-x-x^2-x^3)). - Geoffrey Critzer, Jan 29 2009
a(n) = 2 * a(n-1) + 2^(n-4) - a(n-4) since we can add T or H to a sequence of n-1 flips which has HHH, and H to one which ends in THH and does not have HHH among the first (n-4) flips. - Toby Gottfried, Nov 20 2010
a(n) = 3*a(n-1) - a(n-2) - a(n-3) - 2*a(n-4), a(0)=0, a(1)=0, a(2)=1, a(3)=3. - David Nacin, Mar 07 2012
MATHEMATICA
LinearRecurrence[{3, -1, -1, -2}, {0, 0, 1, 3}, 50] (* David Nacin, Mar 07 2012 *)
PROG
(Python)
def a(n, adict={0:0, 1:0, 2:1, 3:3}):
if n in adict:
return adict[n]
adict[n]=3*a(n-1)-a(n-2)-a(n-3)-2*a(n-4)
return adict[n] # David Nacin, Mar 07 2012
(PARI) concat([0, 0], Vec(1/(1-2*x)/(1-x-x^2-x^3)+O(x^99))) \\ Charles R Greathouse IV, Feb 03 2015
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved