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 A348447 #30 Jan 01 2024 16:06:14
%S A348447 1,1,1,1,2,1,1,3,1,4,4,1,5,5,1,6,9,1,7,14,7,1,8,20,16,4,1,9,27,30,9,1,
%T A348447 10,35,50,25,1,11,44,77,55,11,1,12,54,112,105,36,4,1,13,65,156,182,91,
%U A348447 13,1,14,77,210,294,196,49,1,15,90,275,450,378,140,15
%N A348447 Irregular triangle read by rows: T(n,k) (n >= 0) is the number of ways of selecting k objects from n objects arranged in a circle with no two selected objects having unit separation (i.e. having exactly one object between them).
%C A348447 For odd n, T(n,k) is the number of independent vertex sets of size k in an n-cycle. For even n, T(n,k) is the number of independent vertex sets of size k in two disjoint (n/2)-cycles. Here, a cycle of length one or two is taken to be a path. - _Andrew Howroyd_, Jan 01 2024
%H A348447 Andrew Howroyd, <a href="/A348447/b348447.txt">Table of n, a(n) for n = 0..2578</a> (rows 0..100)
%H A348447 John Konvalina, <a href="https://doi.org/10.1016/0097-3165(81)90006-6">On the number of combinations without unit separation.</a>, Journal of Combinatorial Theory, Series A 31.2 (1981): 101-107. See Table II (which erroneously lacks the n=k=2 element).
%F A348447 G.f.: -3 + y*x + y*(2 + y)*x^2 + (4 - 3*x - y*x^3)/((1 + y*x^2)*(1 - x - y*x^2)). - _Andrew Howroyd_, Jan 01 2024
%e A348447 Triangle begins:
%e A348447 1;
%e A348447 1, 1;
%e A348447 1, 2, 1;
%e A348447 1, 3;
%e A348447 1, 4, 4;
%e A348447 1, 5, 5;
%e A348447 1, 6, 9;
%e A348447 1, 7, 14, 7;
%e A348447 1, 8, 20, 16, 4;
%e A348447 1, 9, 27, 30, 9;
%e A348447 1, 10, 35, 50, 25;
%e A348447 1, 11, 44, 77, 55, 11;
%e A348447 1, 12, 54, 112, 105, 36, 4;
%e A348447 1, 13, 65, 156, 182, 91, 13;
%e A348447 ...
%e A348447 If n = 2 and we call the objects 0 and 1, the permitted sets of objects are {}, {0}, {1}, and {0,1}. If n = 3 and we call the objects 0, 1, and 2, then the permitted sets of objects are {}, {0}, {1}, and {2}; {0,1} is not a permitted set in this case since the objects lie in a circle and 2 lies between 0 and 1 in one direction. - _Michael A. Allen_, Apr 25 2022
%o A348447 (PARI) T(n) = {[Vecrev(p) | p <- Vec(-3 + y*x + y*(2 + y)*x^2 + (4 - 3*x - y*x^3)/((1 + y*x^2)*(1 - x - y*x^2)) + O(x*x^n))]}
%o A348447 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 01 2024
%Y A348447 See A348445 for the case when the n objects are on a line.
%Y A348447 The triangles A034807, A061896, A152060 are very similar.
%Y A348447 The k=2 column is (essentially) A347553.
%K A348447 nonn,tabf
%O A348447 0,5
%A A348447 _N. J. A. Sloane_, Oct 23 2021
%E A348447 Corrected by _Michael A. Allen_, Apr 25 2022
%E A348447 Terms a(56) and beyond from _Andrew Howroyd_, Jan 01 2024