OFFSET
0,2
COMMENTS
With formula a(n) = (5*6^n + 0^n)/6, this is the binomial transform of A083425. - Paul Barry, Apr 30 2003
For n>=1, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
a(n) = (n+1) terms in the sequence (1, 4, 5, 5, 5, ...) dot (n+1) terms in the sequence (1, 1, 5, 30, 180, 1080, ...). Example: a(4) = (1, 4, 5, 5, 5) dot (1, 1, 5, 30, 180) = (1 + 4 + 25 + 150 + 900), where (1, 4, 25, 150, ...) = first differences of current sequence. - Gary W. Adamson, Aug 03 2010
a(n) is the number of compositions of n when there are 5 types of each natural number. - Milan Janjic, Aug 13 2010
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 922
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (6).
FORMULA
a(n) = 6*a(n-1), n>=2.
a(n) = 5*6^(n-1), n>=1. - Vincenzo Librandi, Sep 15 2011
G.f.: (1-x)/(1-6*x).
G.f.: 1/(1 - 5*Sum_{k>=1} x^k).
E.g.f.: (1/6)*(1 + 5*exp(6*x)). - Stefano Spezia, Oct 18 2019
MAPLE
spec := [S, {S=Sequence(Prod(Sequence(Z), Union(Z, Z, Z, Z, Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq(`if`(n=0, 1, 5*6^(n-1)), n=0..30); # G. C. Greubel, Oct 18 2019
MATHEMATICA
Join[{1}, NestList[6#&, 5, 20]] (* Harvey P. Dale, Nov 30 2015 *)
PROG
(PARI) vector(31, n, if(n==1, 1, 5*6^(n-2))) \\ G. C. Greubel, Oct 18 2019
(Magma) [1] cat [5*6^(n-1): n in [1..30]]; // G. C. Greubel, Oct 18 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 22); Coefficients(R!( (1-x)/(1-6*x))); // Marius A. Burtea, Oct 18 2019
(Sage) [1]+[5*6^(n-1) for n in (1..30)] # G. C. Greubel, Oct 18 2019
(GAP) Concatenation([1], List([1..30], n-> 5*6^(n-1) )); # G. C. Greubel, Oct 18 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved