A375978 Number of subsets of {1,2,...,n} such that no two elements differ by 3 or 5.

1, 2, 4, 8, 12, 18, 24, 34, 47, 73, 111, 177, 267, 409, 600, 900, 1324, 2004, 2996, 4564, 6848, 10377, 15513, 23385, 34953, 52685, 78969, 119138, 178840, 269604, 404656, 609310, 914548, 1376530, 2067231, 3111457, 4674751, 7034897, 10570855, 15903377, 23898528

Offset: 0

Author

Michael A. Allen, Sep 20 2024

Keywords

• easy
• nonn

Examples

			For n = 6, the 24 subsets are {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {2,4}, {3,4}, {2,3,4}, {5}, {1,5}, {3,5}, {1,3,5}, {4,5}, {3,4,5}, {6}, {2,6}, {4,6}, {2,4,6}, {5,6}, {4,5,6}.
		

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 (1,1,-1,1,-1,1,1,1,-1,-1,-1,-1).
• Index entries for subsets of {1,..,n} with disallowed differences

Crossrefs

Column k=20 of A376033.

Programs

Mathematica

CoefficientList[Series[(1 + x + x^2 + 3*x^3 + x^4 + x^5 - x^6 - 3*x^7 - 4*x^8 - 3*x^9 - 2*x^10 - x^11)/(1 - x - x^2 + x^3 - x^4 + x^5 - x^6 - x^7 - x^8 + x^9 + x^10 + x^11 + x^12),{x,0,38}],x]
LinearRecurrence[{1, 1, -1, 1, -1, 1, 1, 1, -1, -1, -1, -1}, {1, 2, 4, 8, 12, 18, 24, 34, 47, 73, 111, 177}, 39]

Formula

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) + a(n-6) + a(n-7) + a(n-8) - a(n-9) - a(n-10) - a(n-11) - a(n-12) for n >= 12.

G.f.: (1 + x + x^2 + 3*x^3 + x^4 + x^5 - x^6 - 3*x^7 - 4*x^8 - 3*x^9 - 2*x^10 - x^11)/(1 - x - x^2 + x^3 - x^4 + x^5 - x^6 - x^7 - x^8 + x^9 + x^10 + x^11 + x^12).