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 A262045 #35 Nov 16 2024 02:02:05 %S A262045 1,1,1,1,1,0,0,1,1,1,1,1,1,0,0,1,1,1,2,2,1,1,1,0,0,0,0,1,1,1,1,1,1,1, %T A262045 1,0,1,1,0,1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,2,2,2,2,1,1,1,0,0,0, %U A262045 0,0,0,1,1,1,1,0,0,1,1,1,1,0,1,1,2,2,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1 %N A262045 Irregular triangle read by rows in which row n lists the elements of row n of A249223 and then the elements of the same row in reverse order. %C A262045 The n-th row of the triangle has length 2*A003056(n). %C A262045 This sequence extends A249223 in the same manner as A237593 extends A237591. %C A262045 The entries in the n-th row of the triangle are the widths of the regions between the (n-1)-st and n-th Dyck paths for the symmetric representation of sigma(n) with each column representing the corresponding leg of the n-th path. %H A262045 N. J. A. Sloane, <a href="/A262045/b262045.txt">Table of n, a(n) for n = 1..3200</a> [First 150 rows, based on G. C. Greubel's b-file for A249223] %F A262045 T(n, k) = T(n, 2*A003056(n) + 1 - k) = A249223(n, k), for 1 <= n and 1 <= k <= A003056(n). %e A262045 n\k 1 2 3 4 5 6 7 8 9 10 %e A262045 1 1 1 %e A262045 2 1 1 %e A262045 3 1 0 0 1 %e A262045 4 1 1 1 1 %e A262045 5 1 0 0 1 %e A262045 6 1 1 2 2 1 1 %e A262045 7 1 0 0 0 0 1 %e A262045 8 1 1 1 1 1 1 %e A262045 9 1 0 1 1 0 1 %e A262045 10 1 1 1 0 0 1 1 1 %e A262045 11 1 0 0 0 0 0 0 1 %e A262045 12 1 1 2 2 2 2 1 1 %e A262045 13 1 0 0 0 0 0 0 1 %e A262045 14 1 1 1 0 0 1 1 1 %e A262045 15 1 0 1 1 2 2 1 1 0 1 %e A262045 16 1 1 1 1 1 1 1 1 1 1 %e A262045 17 1 0 0 0 0 0 0 0 0 1 %e A262045 18 1 1 2 1 1 1 1 2 1 1 %e A262045 19 1 0 0 0 0 0 0 0 0 1 %e A262045 20 1 1 1 1 2 2 1 1 1 1 %e A262045 ... %e A262045 The triangle shows that the region between a Dyck path for n and n-1 has width 1 if n is a power of 2. For n a prime the region is a horizontal rectangle of width (height) 1 and the vertical rectangle of width 1 which is its reflection. The Dyck paths and regions are shown below for n = 1..5 (see the A237593 example for n = 1..28): %e A262045 _ _ _ %e A262045 5 |_ _ _| %e A262045 4 |_ _ |_ _ %e A262045 3 |_ _|_ | | %e A262045 2 |_ | | | | %e A262045 1 |_|_|_|_|_| %t A262045 (* functions a237048[ ] and row[ ] are defined in A237048 *) %t A262045 f[n_] :=Drop[FoldList[Plus, 0, Map[(-1)^(#+1)&, Range[row[n]]] a237048[n]], 1] %t A262045 a262045[n_]:=Join[f[n], Reverse[f[n]]] %t A262045 Flatten[Map[a262045, Range[16]]](* data *) %Y A262045 Cf. A000203, A003056, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A249223, A262048. %K A262045 nonn,tabf %O A262045 1,19 %A A262045 _Hartmut F. W. Hoft_, Sep 09 2015