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 A380231 #75 May 30 2025 23:59:17 %S A380231 1,2,1,2,1,4,3,4,5,4,3,6,5,4,7,8,7,8,7,10,9,8,7,10,11,10,9,12,11,14, %T A380231 13,14,13,12,15,16,15,14,13,16,15,18,17,16,19,18,17,20,21,22,21,20,19, %U A380231 22,21,24,23,22,21,24,23,22,25,26,25,28,27,26,25,28,27,32,31,30,29,28,31,30,29 %N A380231 Alternating row sums of triangle A237591. %C A380231 Consider the symmetric Dyck path in the first quadrant of the square grid described in the n-th row of A237593. Let C = (A240542(n), A240542(n)) be the middle point of the Dyck path. %C A380231 a(n) is also the coordinate on the x axis of the point (a(n),n) and also the coordinate on the y axis of the point (n,a(n)) such that the middle point of the line segment [(a(n),n),(n,a(n))] coincides with the middle point C of the symmetric Dyck path. %C A380231 The three line segments [(a(n),n),C], [(n,a(n)),C] and [(n,n),C] have the same length. %C A380231 For n > 2 the points (n,n), C and (a(n),n) are the vertices of a virtual isosceles right triangle. %C A380231 For n > 2 the points (n,n), C and (n,a(n)) are the vertices of a virtual isosceles right triangle. %C A380231 For n > 2 the points (a(n),n), (n,n) and (n,a(n)) are the vertices of a virtual isosceles right triangle. %F A380231 a(n) = 2*A240542(n) - n. %F A380231 a(n) = n - 2*A322141(n). %F A380231 a(n) = A240542(n) - A322141(n). %e A380231 For n = 14 the 14th row of A237591 is [8, 3, 1, 2] hence the alternating row sum is 8 - 3 + 1 - 2 = 4, so a(14) = 4. %e A380231 On the other hand the 14th row of A237593 is the 14th row of A237591 together with the 14 th row of A237591 in reverse order as follows: [8, 3, 1, 2, 2, 1, 3, 8]. %e A380231 Then with the terms of the 14th row of A237593 we can draw a Dyck path in the first quadrant of the square grid as shown below: %e A380231 . %e A380231 (y axis) %e A380231 . %e A380231 . %e A380231 . (4,14) (14,14) %e A380231 ._ _ _ . _ _ _ _ . %e A380231 . | %e A380231 . | %e A380231 . |_ %e A380231 . | %e A380231 . |_ _ %e A380231 . C |_ _ _ %e A380231 . | %e A380231 . | %e A380231 . | %e A380231 . | %e A380231 . . (14,4) %e A380231 . | %e A380231 . | %e A380231 . . . . . . . . . . . . . . | . . . (x axis) %e A380231 (0,0) %e A380231 . %e A380231 In the example the point C is the point (9,9). %e A380231 The three line segments [(4,14),(9,9)], [(14,4),(9,9)] and [(14,14),(9,9)] have the same length. %e A380231 The points (14,14), (9,9) and (4,14) are the vertices of a virtual isosceles right triangle. %e A380231 The points (14,14), (9,9) and (14,4) are the vertices of a virtual isosceles right triangle. %e A380231 The points (4,14), (14,14) and (14,4) are the vertices of a virtual isosceles right triangle. %o A380231 (PARI) row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i); %o A380231 a(n) = my(orow = concat(row235791(n), 0)); vecsum(vector(#orow-1, i, (-1)^(i+1)*(orow[i] - orow[i+1]))); \\ _Michel Marcus_, Apr 13 2025 %Y A380231 Other alternating row sums (ARS) related to the Dyck paths of A237593 and the stepped pyramid described in A245092 are as follows: %Y A380231 ARS of A237593 give A000004. %Y A380231 ARS of A196020 give A000203. %Y A380231 ARS of A252117 give A000203. %Y A380231 ARS of A271343 give A000593. %Y A380231 ARS of A231347 give A001065. %Y A380231 ARS of A236112 give A004125. %Y A380231 ARS of A236104 give A024916. %Y A380231 ARS of A249120 give A024916. %Y A380231 ARS of A271344 give A033879. %Y A380231 ARS of A231345 give A033880. %Y A380231 ARS of A239313 give A048050. %Y A380231 ARS of A237048 give A067742. %Y A380231 ARS of A236106 give A074400. %Y A380231 ARS of A235794 give A120444. %Y A380231 ARS of A266537 give A146076. %Y A380231 ARS of A236540 give A153485. %Y A380231 ARS of A262612 give A175254. %Y A380231 ARS of A353690 give A175254. %Y A380231 ARS of A239446 give A235796. %Y A380231 ARS of A239662 give A239050. %Y A380231 ARS of A235791 give A240542. %Y A380231 ARS of A272026 give A272027. %Y A380231 ARS of A211343 give A336305. %Y A380231 Cf. A221529, A237270, A237271, A237591, A262626, A322141. %K A380231 nonn %O A380231 1,2 %A A380231 _Omar E. Pol_, Jan 17 2025