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A375977 Number of subsets of {1,2,...,n} such that no two elements differ by 2 or 5.

%I A375977 #13 Mar 19 2025 08:34:17
%S A375977 1,2,4,6,9,15,21,29,45,69,100,152,236,349,517,789,1185,1757,2653,4014,
%T A375977 5992,8986,13573,20363,30485,45901,69041,103481,155468,233908,351104,
%U A375977 527033,792405,1190493,1787129,2685209,4035261,6059758,9101828,13676670,20544125
%N A375977 Number of subsets of {1,2,...,n} such that no two elements differ by 2 or 5.
%H A375977 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 A375977 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 A375977 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,-1,1,2,-1,0,1,-1).
%H A375977 <a href="/index/Su#sublatts">Index entries for subsets of {1,..,n} with disallowed differences</a>
%F A375977 a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) + 2*a(n-7) - a(n-8) + a(n-10) - a(n-11) for n >= 11.
%F A375977 G.f.: (1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 - x^7 - x^10)/(1 - x - x^3 + x^5 - x^6 - 2*x^7 + x^8 - x^10 + x^11).
%e A375977 For n = 6, the 21 subsets are {}, {1}, {2}, {1,2}, {3}, {2,3}, {4}, {1,4}, {3,4}, {5}, {1,5}, {2,5}, {1,2,5}, {4,5}, {1,4,5}, {6}, {2,6}, {3,6}, {2,3,6}, {5,6}, {2,5,6}.
%t A375977 CoefficientList[Series[(1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 - x^7 - x^10)/(1 - x - x^3 + x^5 - x^6 - 2*x^7 + x^8 - x^10 + x^11),{x,0,39}],x]
%t A375977 LinearRecurrence[{1, 0, 1, 0, -1, 1, 2, -1, 0, 1, -1}, {1, 2, 4, 6, 9, 15, 21, 29, 45, 69, 100}, 39]
%Y A375977 Column k=18 of A376033.
%K A375977 nonn,easy
%O A375977 0,2
%A A375977 _Michael A. Allen_, Sep 20 2024