OFFSET
1,6
COMMENTS
a(n-1) is the number of compositions of n with at least one part >= 6. - 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
Félix Balado and Guénolé C. M. Silvestre, Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings, arXiv:2602.10005 [math.CO], 2026. See p. 20.
Eric Weisstein's World of Mathematics, Run.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-1,-1,-2).
FORMULA
a(n) = 2^(n+1) - pentanacci(n+6), cf. A001591. - Vladeta Jovovic, Feb 23 2003
G.f.: x^5/((1-2*x)*(1-x-x^2-x^3-x^4-x^5)). - Geoffrey Critzer, Jan 29 2009
a(n) = 3*a(n-1) - a(n-2) - a(n-3) - a(n-4) - a(n-5) - 2*a(n-6). - Wesley Ivan Hurt, Jan 03 2021
MATHEMATICA
f[x_] := x^4 / (1-3x+x^2+x^3+x^4+x^5+2x^6); CoefficientList[ Series[f[x], {x, 0, 31}], x] (* Jean-François Alcover, Nov 18 2011 *)
LinearRecurrence[{3, -1, -1, -1, -1, -2}, {0, 0, 0, 0, 1, 3}, 40] (* Harvey P. Dale, Jan 27 2015 *)
PROG
(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; -2, -1, -1, -1, -1, 3]^(n-1)*[0; 0; 0; 0; 1; 3])[1, 1] \\ Charles R Greathouse IV, Jun 15 2015
(Magma)
R<x>:= PowerSeriesRing(Integers(), 50);
[0, 0, 0, 0] cat Coefficients(R!( x^5/((1-2*x)*(1-x-x^2-x^3-x^4-x^5)) )); // G. C. Greubel, Jun 01 2025
(SageMath)
def A050233_list(prec):
P.<x>= PowerSeriesRing(QQ, prec)
return P( x^5/((1-2*x)*(1-x-x^2-x^3-x^4-x^5)) ).list()
a=A050233_list(41); a[1:] # G. C. Greubel, Jun 01 2025
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved
