A375979 Number of subsets of {1,2,...,n} such that no two elements differ by 4 or 5.

1, 2, 4, 8, 16, 24, 32, 42, 55, 76, 118, 192, 314, 504, 767, 1120, 1612, 2324, 3412, 5148, 7900, 12169, 18631, 28152, 42024, 62364, 92576, 138141, 207629, 313718, 474796, 717456, 1080320, 1620994, 2427447, 3634800, 5450293, 8188936, 12323172, 18555880, 27930853

Offset: 0

Author

Michael A. Allen, Sep 20 2024

Keywords

• nonn
• easy

Examples

			For n = 6, the 32 subsets are {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, {4}, {1,4}, {2,4}, {1,2,4}, {3,4}, {1,3,4}, {2,3,4}, {1,2,3,4}, {5}, {2,5}, {3,5}, {2,3,5}, {4,5}, {2,4,5}, {3,4,5}, {2,3,4,5}, {6}, {3,6}, {4,6}, {3,4,6}, {5,6}, {3,5,6}, {4,5,6}, {3,4,5,6}.
		

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 (1,0,1,-1,0,1,1,2,3,-1,-2,-3).
• Index entries for subsets of {1,..,n} with disallowed differences

Crossrefs

Column k=24 of A376033.

Programs

Mathematica

CoefficientList[Series[(1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 6*x^5 + 3*x^6 - x^7 - 3*x^8 - 6*x^9 - 5*x^10 - 3*x^11)/(1 - x - x^3 + x^4 - x^6 - x^7 - 2*x^8 - 3*x^9 + x^10 + 2*x^11 + 3*x^12),{x,0,38}],x]
LinearRecurrence[{1, 0, 1, -1, 0, 1, 1, 2, 3, -1, -2, -3}, {1, 2, 4, 8, 16, 24, 32, 42, 55, 76, 118, 192}, 39]

Formula

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

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