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%I #12 Sep 20 2024 06:23:23
%S 1,2,4,6,8,11,14,18,25,37,53,75,105,145,198,274,383,537,752,1053,1468,
%T 2041,2838,3954,5513,7693,10737,14979,20880,29100,40558,56538,78828,
%U 109926,153289,213736,297991,415448,579198,807519,1125889,1569813,2188752
%N Number of subsets of {1,2,...,n} such that no two elements differ by 2, 3, or 5.
%C a(n) is the number of compositions of n+5 into parts 1, 6, 7, 10, 14, 18, 22, ...
%H Michael A. Allen, <a href="https://doi.org/10.22049/CCO.2024.29370.1959">Combinations without specified separations</a>, Communications in Combinatorics and Optimization (in press).
%H Michael A. Allen, <a href="https://doi.org/10.48550/arXiv.2409.00624">Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings</a>, arXiv:2409.00624 [math.CO], 2024.
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1,1,1,0,0,0,-1).
%F a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) + a(n-7) - a(n-11) for n >= 11.
%F G.f.: (1 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 - x^7 - x^8 - x^9 - x^10)/(1 - x - x^4 + x^5 - x^6 - x^7 + x^11).
%e For n = 6, the 14 subsets are {}, {1}, {2}, {1,2}, {3}, {2,3}, {4}, {3,4}, {5}, {1,5}, {4,5}, {6}, {2,6}, {5,6}.
%t CoefficientList[Series[(1 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 - x^7 - x^8 - x^9 - x^10)/(1 - x - x^4 + x^5 - x^6 - x^7 + x^11),{x,0,42}],x]
%t LinearRecurrence[{1, 0, 0, 1, -1, 1, 1, 0, 0, 0, -1}, {1, 2, 4, 6, 8, 11, 14, 18, 25, 37, 53}, 42]
%Y See A375981 for other sequences related to restricted combinations.
%K easy,nonn
%O 0,2
%A _Michael A. Allen_, Sep 04 2024