This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385870 #6 Jul 15 2025 18:26:33
%S A385870 1,2,3,5,8,13,21,29,45,66,99,148,218,337,497,755,1131,1699,2571,3824,
%T A385870 5794,8661,13041,19601,29376,44311,66349,99936,150000,225387,339000,
%U A385870 508631,765392,1148865,1727249,2595270,3898324,5861084,8801690,13231745,19877092,29869125
%N A385870 Number of subsets of {1,2,...,n} such that no two elements differ by 1 or 6.
%C A385870 a(n) is the number of permutations of 0..n with each element moved by -1 to 1 places and every 6 consecutive elements having their maximum within 6 of their minimum.
%H A385870 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 A385870 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 A385870 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,1,-1,1,-1,0,1,-1,1,-1,1,0,-1,1,-1).
%H A385870 <a href="/index/Su#sublatts">Index entries for subsets of {1,..,n} with disallowed differences</a>
%F A385870 G.f.: (1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 5*x^6 + 3*x^8 + 2*x^9 - x^10 + x^11 - 4*x^12 - x^13 - 2*x^14 - 2*x^15 - x^17)/(1 - x - x^4 - x^5 + 2*x^6 - 4*x^7 + 2*x^8 - 2*x^10 + 2*x^11 - 4*x^12 + 3*x^13 - x^14 + 2*x^16 - x^17 + x^18).
%F A385870 a(n) = a(n-1) + a(n-4) + a(n-5) - 2*a(n-6) + 4*a(n-7) - 2*a(n-8) + 2*a(n-10) - 2*a(n-11) + 4*a(n-12) - 3*a(n-13) + a(n-14) - 2*a(n-16) + a(n-17) - a(n-18) for n >= 18.
%e A385870 For n = 7, the 29 subsets are {}, {1}, {2}, {3}, {1,3}, {4}, {1,4}, {2,4}, {5}, {1,5}, {2,5}, {3,5}, {1,3,5}, {6}, {1,6}, {2,6}, {3,6}, {1,3,6}, {4,6}, {1,4,6}, {2,4,6}, {7}, {2,7}, {3,7}, {4,7}, {2,4,7}, {5,7}, {2,5,7}, {3,5,7}.
%t A385870 CoefficientList[Series[(1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 5*x^6 + 3*x^8 + 2*x^9 - x^10 + x^11 - 4*x^12 - x^13 - 2*x^14 - 2*x^15 - x^17)/(1 - x - x^4 - x^5 + 2*x^6 - 4*x^7 + 2*x^8 - 2*x^10 + 2*x^11 - 4*x^12 + 3*x^13 - x^14 + 2*x^16 - x^17 + x^18),{x,0,41}],x]
%t A385870 LinearRecurrence[{1,0,0,1,1,-2,4,-2,0,2,-2,4,-3,1,0,-2,1,-1},{1,2,3,5,8,13,21,29,45,66,99,148,218,337,497,755,1131,1699}, 42]
%Y A385870 Column k=33 of A376033.
%K A385870 easy,nonn
%O A385870 0,2
%A A385870 _Michael A. Allen_, Jul 11 2025