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 A348445 #28 Jan 01 2024 16:06:19
%S A348445 1,1,1,1,2,1,1,3,2,1,4,4,1,5,7,2,1,6,11,6,1,1,7,16,13,3,1,8,22,24,9,1,
%T A348445 9,29,40,22,3,1,10,37,62,46,12,1,1,11,46,91,86,34,4,1,12,56,128,148,
%U A348445 80,16,1,13,67,174,239,166,50,4,1,14,79,230,367,314,130,20,1,1,15,92,297,541,553,296,70,5
%N A348445 Irregular triangle read by rows: T(n,k) (n >= 0) is the number of ways of selecting k objects from n objects arranged on a line with no two selected objects having unit separation (i.e. having exactly one object between them).
%C A348445 Equivalently, T(n,k) is the number of independent vertex sets of size k in two disjoint paths, one of length floor(n/2) and the other of length ceiling(n/2). - _Andrew Howroyd_, Jan 01 2024
%H A348445 Andrew Howroyd, <a href="/A348445/b348445.txt">Table of n, a(n) for n = 0..2675</a> (rows 0..100)
%H A348445 Kenneth Edwards and Michael A. Allen, <a href="https://arxiv.org/abs/2009.04649">New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile</a>, arXiv:2009.04649 [math.CO], 2020.
%H A348445 Kenneth Edwards and Michael A. Allen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Allen/edwards2.html">New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile</a>, J. Int. Seq. 24 (2021) Article 21.3.8.
%H A348445 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 I.
%F A348445 T(n,k) = A335964(n+2,k).
%F A348445 T(n,0) = 1.
%F A348445 T(n,1) = n.
%F A348445 T(1,k) = 0 if k>1.
%F A348445 T(n,k) = T(n-1,k) + T(n-3,k-1) + T(n-4,k-2).
%F A348445 G.f.: (1 + y*x + (y^2 + y)*x^2 + y^2*x^3)/((1 + y*x^2)*(1 - x - y*x^2)). - _Andrew Howroyd_, Jan 01 2024
%e A348445 Triangle begins:
%e A348445 1;
%e A348445 1, 1;
%e A348445 1, 2, 1;
%e A348445 1, 3, 2;
%e A348445 1, 4, 4;
%e A348445 1, 5, 7, 2;
%e A348445 1, 6, 11, 6, 1;
%e A348445 1, 7, 16, 13, 3;
%e A348445 1, 8, 22, 24, 9;
%e A348445 1, 9, 29, 40, 22, 3;
%e A348445 ...
%t A348445 Flatten[Drop[CoefficientList[CoefficientList[Series[1/((1+x^2*y)(1-x-x^2*y)),{x, 0, 17}],x],y],2]] (* _Michael A. Allen_, Dec 27 2021 *)
%o A348445 (PARI) T(n) = {[Vecrev(p) | p <- Vec((1 + y*x + (y^2 + y)*x^2 + y^2*x^3)/((1 + y*x^2)*(1 - x - y*x^2)) + O(x*x^n))]}
%o A348445 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 01 2024
%Y A348445 Cf. A335964.
%Y A348445 See A348447 for the circular case.
%K A348445 nonn,tabf
%O A348445 0,5
%A A348445 _N. J. A. Sloane_, Oct 22 2021
%E A348445 Terms corrected and extended by _Michael A. Allen_, Dec 27 2021