OFFSET
0,5
COMMENTS
Number of compositions (ordered partitions) of n into elements of the set {1,4,5}.
a(n+3) is the number of length-n binary words with no substring 1x1 of 1xy1 (that is, no 1's occur with distance two or three), see fxtbook link. - Joerg Arndt, May 27 2011
REFERENCES
D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..5708
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See p. 18.
Said Amrouche, Hacène Belbachir, Unimodality and linear recurrences associated with rays in the Delannoy triangle, Turkish Journal of Mathematics (2019) Vol. 44, 118-130.
Joerg Arndt, Matters Computational (The Fxtbook), section 14.10.3, p. 322
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,1).
FORMULA
a(n) = a(n-1) + a(n-4) + a(n-5).
G.f.: 1/(1-x-x^4-x^5).
a(n) = Sum_{k=0..n} Sum_{j=floor((n-k)/4)..floor((n-k)/3)} binomial(k,j)*binomial(j,n-k-3*j). - Vladimir Kruchinin, May 26 2011
MATHEMATICA
CoefficientList[Series[1/(1 - x - x^4 - x^5), {x, 0, 41}], x] (* Michael De Vlieger, Feb 03 2020 *)
PROG
(Maxima)
a(n):=sum(sum(binomial(k, j)*binomial(j, n-k-3*j), j, floor((n-k)/4), floor((n-k)/3)), k, 0, n); /* Vladimir Kruchinin, May 26 2011 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Baltic, Feb 17 2003
STATUS
approved