OFFSET
0,2
COMMENTS
a(n) is the number of compositions of n+5 into parts 1, 6, 8, 9, 12, 15, 18, 21, ...
LINKS
Michael A. Allen, Combinations without specified separations, Communications in Combinatorics and Optimization (in press).
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,1,0,1,0,0,-1).
FORMULA
a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-6) + a(n-8) - a(n-11) for n >= 11.
G.f.: (1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 - x^8 - x^9 - x^10)/(1 - x - x^3 + x^4 - x^6 - x^8 + x^11).
EXAMPLE
For n = 6, the 14 subsets are {}, {1}, {2}, {3}, {1,3}, {4}, {1,4}, {2,4}, {5}, {2,5}, {3,5}, {6}, {3,6}, {4,6}.
The a(4) = 8 compositions of 9 into parts 1, 6, 8, 9, ... are 1+1+1+1+1+1+1+1+1, 1+1+1+6, 1+1+6+1, 1+6+1+1, 6+1+1+1, 1+8, 8+1, 9.
MATHEMATICA
CoefficientList[Series[(1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 - x^8 - x^9 - x^10)/(1 - x - x^3 + x^4 - x^6 - x^8 + x^11), {x, 0, 42}], x]
LinearRecurrence[{1, 0, 1, -1, 0, 1, 0, 1, 0, 0, -1}, {1, 2, 3, 5, 8, 11, 14, 19, 25, 34, 49}, 42]
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Michael A. Allen, Sep 04 2024
STATUS
approved