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 A187660 #48 Jan 10 2022 03:29:08
%S A187660 1,1,-1,1,-1,-1,1,-2,-1,1,1,-2,-3,1,1,1,-3,-3,4,1,-1,1,-3,-6,4,5,-1,
%T A187660 -1,1,-4,-6,10,5,-6,-1,1,1,-4,-10,10,15,-6,-7,1,1,1,-5,-10,20,15,-21,
%U A187660 -7,8,1,-1,1,-5,-15,20,35,-21,-28,8,9,-1,-1,1,-6,-15,35,35,-56,-28,36,9,-10,-1,1
%N A187660 Triangle read by rows: T(n,k) = (-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k), 0 <= k <= n.
%C A187660 Conjecture: (i) Let n > 1 and N=2*n+1. Row n of T gives the coefficients of the characteristic polynomial p_N(x)=Sum_{k=0..n} T(n,k)*x^(n-k) of the n X n Danzer matrix D_{N,n-1} = {{0,...,0,1}, {0,...,0,1,1}, ..., {0,1,...,1}, {1,...,1}}. (ii) Let S_0(t)=1, S_1(t)=t and S_r(t)=t*S_(r-1)(t)-S_(r-2)(t), r > 1 (cf. A049310). Then p_N(x)=0 has solutions w_{N,j}=S_(n-1)(phi_{N,j}), where phi_{N,j}=2*(-1)^(j+1)*cos(j*Pi/N), j = 1..n. - _L. Edson Jeffery_, Dec 18 2011
%H A187660 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Danzer matrices</a>
%H A187660 Guoce Xin and Yueming Zhong, <a href="https://arxiv.org/abs/2201.02376">Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials</a>, arXiv:2201.02376 [math.CO], 2022.
%F A187660 T(n,k) = (-1)^n*A066170(n,k).
%F A187660 abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) = abs(A130777(n,k)).
%F A187660 abs(T(n,k)) = A065941(n,n-k) = abs(A108299(n,n-k)).
%e A187660 Triangle begins:
%e A187660 1;
%e A187660 1, -1;
%e A187660 1, -1, -1;
%e A187660 1, -2, -1, 1;
%e A187660 1, -2, -3, 1, 1;
%e A187660 1, -3, -3, 4, 1, -1;
%e A187660 1, -3, -6, 4, 5, -1, -1;
%e A187660 1, -4, -6, 10, 5, -6, -1, 1;
%e A187660 1, -4, -10, 10, 15, -6, -7, 1, 1;
%e A187660 1, -5, -10, 20, 15, -21, -7, 8, 1, -1;
%e A187660 1, -5, -15, 20, 35, -21, -28, 8, 9, -1, -1;
%e A187660 1, -6, -15, 35, 35, -56, -28, 36, 9, -10, -1, 1;
%p A187660 A187660 := proc(n,k): (-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k) end: seq(seq(A187660(n,k), k=0..n), n=0..11); # _Johannes W. Meijer_, Aug 08 2011
%t A187660 t[n_, k_] := (-1)^Floor[3 k/2] Binomial[Floor[(n + k)/2], k]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] (* _L. Edson Jeffery_, Oct 20 2017 *)
%Y A187660 Signed version of A046854.
%Y A187660 Absolute values of a(n) form a reflected version of A065941, which is considered the main entry.
%Y A187660 Cf. A046854, A066170, A130777, A267482.
%K A187660 sign,easy,tabl
%O A187660 0,8
%A A187660 _L. Edson Jeffery_, Mar 12 2011
%E A187660 Edited and corrected by _L. Edson Jeffery_, Oct 20 2017