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 A342344 #76 Mar 18 2021 23:59:29 %S A342344 0,0,2,3,1,3,1,2,1,2,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1,1,2,1,2,1,2,1,1, %T A342344 1,2,1,1,1,2,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,2,1,1, %U A342344 1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,2 %N A342344 Number of parts in the symmetric representation of antisigma(n). %C A342344 In order to construct this sequence and the diagram of the symmetric representation of antisigma(n) = A024816(n) we use the following rules: %C A342344 At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593. The area of the region that is below the symmetric representation of sigma(n) equals A024916(n-1). %C A342344 At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n). Then we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals A244048(n) = A153485(n-1). The area of the region that is below the staircase equals A000217(n-1). %C A342344 At stage 3 we turn OFF the cells of the symmetric representation of sigma(n) and also the cells that are below the staircase. Then we turn ON the rest of the cells that are in the square n X n. The result is that the ON cell form the diagram of the symmetric representation of antisigma(n) = A024816(n). See the Example section. %C A342344 For n >= 7; if A237271(n) = 1 or n is a term of A262259 then a(n) = 2 otherwise a(n) = 1. %e A342344 Illustration of the symmetric representation of antisigma(n) = AS(n) = A024816(n), for n = 1..6: %e A342344 . y| _ _ %e A342344 . y| _ _ | _ _ |_ | %e A342344 . y| _ | _ _| | | |_ | |_| %e A342344 . y| _ | _ |_| | |_ _| | |_|_ _ %e A342344 . y| | _|_| | |_|_ | |_ | | |_ | %e A342344 . y| | | |_| | |_| | |_| | |_| %e A342344 . |_ _ |_ _ _ |_ _ _ _ |_ _ _ _ _ |_ _ _ _ _ _ |_ _ _ _ _ _ _ %e A342344 . x x x x x x %e A342344 . %e A342344 n: 1 2 3 4 5 6 %e A342344 a(n): 0 0 2 3 1 3 %e A342344 AS(n): 0 0 2 3 9 9 %e A342344 . %e A342344 Illustration of the symmetric representation of antisigma(n) = AS(n) = A024816(n), for n = 7..9: %e A342344 . y| _ _ _ _ %e A342344 . y| _ _ _ | _ _ _ _| | %e A342344 . y| _ _ _ | _ _ _ | | | |_ _ _ | %e A342344 . | _ _ _| | | |_ | |_ | | |_ |_ | | %e A342344 . | |_ | | |_ |_ |_ _| | |_ |_| _| %e A342344 . | |_ _| | |_ |_ _ | |_ | %e A342344 . | |_ | | |_ | | |_ | %e A342344 . | |_ | | |_ | | |_ | %e A342344 . | |_| | |_| | |_| %e A342344 . |_ _ _ _ _ _ _ _ |_ _ _ _ _ _ _ _ _ |_ _ _ _ _ _ _ _ _ _ %e A342344 . x x x %e A342344 . %e A342344 n: 7 8 9 %e A342344 a(n): 1 2 1 %e A342344 AS(n): 20 21 32 %e A342344 . %e A342344 For n = 9 the figures 1, 2 and 3 below show respectively the three stages described in the Comments section as follows: %e A342344 . %e A342344 . y|_ _ _ _ _ 5 y|_ _ _ _ _ _ _ _ _ y| _ _ _ _ %e A342344 . |_ _ _ _ _| |_ _ _ _ _| | | _ _ _ _| | %e A342344 . | |_ _ 3 | |_ |_ _ R | | |_ _ _ | %e A342344 . | |_ | | |_ |_ | | | |_ |_ | | %e A342344 . | |_|_ _ 5 | |_ T |_|_ _| | |_ |_| _| %e A342344 . | | | | |_ | | | |_ | %e A342344 . | Q | | | |_ | | | |_ | %e A342344 . | | | | W |_ | | | |_ | %e A342344 . | | | | |_| | | |_| %e A342344 . |_ _ _ _ _ _ _ _|_|_ |_ _ _ _ _ _ _ _|_|_ |_ _ _ _ _ _ _ _ _ _ %e A342344 . x x x %e A342344 . Figure 1. Figure 2. Figure 3. %e A342344 . Symmetric Symmetric Symmetric %e A342344 . representation representation representation %e A342344 . of sigma(9) of sigma(9) of antisigma(9) %e A342344 . A000203(9) = 13 A000203(9) = 13 A024816(9) = 32 %e A342344 . and of and of %e A342344 . Q = A024916(8) = 56 R = A004125(9) = 12 %e A342344 . T = A244048(9) = 20 %e A342344 . T = A153485(8) = 20 %e A342344 . W = A000217(8) = 36 %e A342344 . %e A342344 Note that the symmetric representation of antisigma(9) contains a hole formed by three cells because these three cells were the central part of the symmetric representation of sigma(9). %Y A342344 Cf. A000203, A000217, A000290, A004125, A024816, A024916, A153485, A174973, A236104, A237270, A237271, A237593, A238443, A239660, A239931, A239932, A239933, A239934, A244048, A262259. %K A342344 nonn %O A342344 1,3 %A A342344 _Omar E. Pol_, Mar 08 2021