A349843 Expansion of (1 - x^2)/((1 - x^10)*(1 - x - x^2)).

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 56, 90, 145, 235, 380, 615, 995, 1610, 2605, 4215, 6821, 11036, 17856, 28892, 46748, 75640, 122388, 198028, 320416, 518444, 838861, 1357305, 2196165, 3553470, 5749635, 9303105

Offset: 0

Author

Michael A. Allen, Dec 13 2021

Keywords

• easy
• nonn

Comments

The number of compositions of n using elements from the set {1,3,5,7,9,10}.

Number of ways to tile an n-board (an n X 1 array of 1 X 1 cells) using squares, trominoes, pentominoes, heptominoes, nonominoes, and decominoes.

Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-1,0,2,4,6,8,9} for all i=1,...,n.

a(n) gives the sums of the antidiagonals of A349841.

References

• D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Links

• Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
• K. Edwards and M. A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,0,1,0,1,1).

Crossrefs

Sums of antidiagonals of triangles in the same family as A349841: A000045, A006498, A079962, A349840.

Programs

Mathematica

CoefficientList[Series[1/(1-x-x^3-x^5-x^7-x^9-x^10), {x, 0, 35}], x]

Formula

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) + a(n-9) + a(n-10) + delta(n,0), a(n<0)=0.

a(n) = a(n-1) + a(n-2) + a(n-10) - a(n-11) - a(n-12) + delta(n,0) - delta(n,2), a(n<0)=0.

G.f.: 1/(1-x-x^3-x^5-x^7-x^9-x^10).