A281010 - cp's OEIS frontend

A281010 Triangle read by rows in which row 2n-1 lists the widths of the symmetric representation of sigma(n), and row 2n lists a finite sequence S together with -1, with the property that the partial sums of S give the row 2n-1.

%I A281010 #31 Apr 18 2017 16:32:41
%S A281010 1,1,-1,1,1,1,1,0,0,-1,1,1,0,1,1,1,0,-1,1,0,-1,1,1,1,1,1,1,1,1,0,0,0,
%T A281010 0,0,0,-1,1,1,1,0,0,0,1,1,1,1,0,0,-1,0,0,1,0,0,-1,1,1,1,1,1,2,1,1,1,1,
%U A281010 1,1,0,0,0,0,1,-1,0,0,0,0,-1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,-1,0,0,0,0,1,0,0,0,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1
%N A281010 Triangle read by rows in which row 2n-1 lists the widths of the symmetric representation of sigma(n), and row 2n lists a finite sequence S together with -1, with the property that the partial sums of S give the row 2n-1.
%C A281010 The row 2n-1 lists the widths of the terraces at the n-th level (starting from the top) of the pyramid described in A245092.
%C A281010 The sum of the areas of these terraces equals A000203(n): the sum of the divisors of n.
%C A281010 The k-th element of row 2n is associated to the k-th vertical cells at the n-th level of the pyramid.
%C A281010 The row 2n shows where the subparts (or subregions) of the terraces starting and ending, in accordance with the values 1 or -1.
%C A281010 The number of subparts in the n-th terrace equals A001227(n): the number of odd divisors of n.
%C A281010 If n is odd then the number of subparts in the n-th terrace is also A000005(n): the number of divisors of n.
%H A281010 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)</a>
%H A281010 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the pyramid (first 16 levels)</a>
%e A281010 Triangle begins:
%e A281010 1;
%e A281010 1,-1;
%e A281010 1, 1, 1;
%e A281010 1, 0, 0,-1;
%e A281010 1, 1, 0, 1, 1;
%e A281010 1, 0,-1, 1, 0;-1;
%e A281010 1, 1, 1, 1, 1, 1, 1;
%e A281010 1, 0, 0, 0, 0, 0, 0,-1;
%e A281010 1, 1, 1, 0, 0, 0, 1, 1, 1;
%e A281010 1, 0, 0,-1, 0, 0, 1, 0, 0,-1;
%e A281010 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1;
%e A281010 1, 0, 0, 0, 0, 1,-1, 0, 0, 0, 0,-1;
%e A281010 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1;
%e A281010 1, 0, 0, 0,-1, 0, 0, 0, 0, 1, 0, 0, 0,-1;
%e A281010 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A281010 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1;
%e A281010 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1;
%e A281010 1, 0, 0, 0, 0,-1, 0, 1, 0, 0,-1, 0, 1, 0, 0, 0, 0,-1;
%e A281010 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A281010 1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 0, 0, 0, 0, 0, 0,-1;
%e A281010 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1;
%e A281010 1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,-1;
%e A281010 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A281010 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0,-1;
%e A281010 ...
%e A281010 Written as an isosceles triangle the sequence begins:
%e A281010 .
%e A281010 .                                        1;
%e A281010 .                                      1, -1;
%e A281010 .                                    1,  1,  1;
%e A281010 .                                  1,  0,  0, -1;
%e A281010 .                                1,  1,  0,  1,  1;
%e A281010 .                              1,  0, -1,  1,  0, -1;
%e A281010 .                            1,  1,  1,  1,  1,  1,  1;
%e A281010 .                          1,  0,  0,  0,  0,  0,  0, -1;
%e A281010 .                        1,  1,  1,  0,  0,  0,  1,  1,  1;
%e A281010 .                      1,  0,  0, -1,  0,  0,  1,  0,  0, -1;
%e A281010 .                    1,  1,  1,  1,  1,  2,  1,  1,  1,  1,  1;
%e A281010 .                  1,  0,  0,  0,  0,  1, -1,  0,  0,  0,  0, -1;
%e A281010 .                1,  1,  1,  1,  0,  0,  0,  0,  0,  1,  1,  1,  1;
%e A281010 .              1,  0,  0,  0, -1,  0,  0,  0,  0,  1,  0,  0,  0, -1;
%e A281010 .            1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1;
%e A281010 .          1,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1;
%e A281010 .        1,  1,  1,  1,  1,  0,  0,  1,  1,  1,  0,  0,  1,  1,  1,  1,  1;
%e A281010 .      1,  0,  0,  0,  0, -1,  0,  1,  0,  0, -1,  0,  1,  0,  0,  0,  0, -1;
%e A281010 .    1,  1,  1,  1,  1,  1,  1,  1,  1,  0,  1,  1,  1,  1,  1,  1,  1,  1,  1;
%e A281010 .  1,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1,  0,  0,  0,  0,  0,  0,  0,  0, -1;
%e A281010 ...
%Y A281010 The sum of row 2n-1 is A000203(n).
%Y A281010 The sum of row 2n is A000004(n) = 0.
%Y A281010 The number of positive terms in row 2n is A001227(n).
%Y A281010 The number of nonzero terms in row 2n is A054844(n).
%Y A281010 Middle diagonal (or central column of the isosceles triangle) gives A067742.
%Y A281010 Row 2n-1 is also the n-th row of A249351.
%Y A281010 Row 2n is also the n-th row of A281011.
%Y A281010 Row 2n-1 lists the partial sums of the terms, except the last term, of the row 2n.
%Y A281010 Cf. A000005, A001227, A196020, A235048, A236104, A237048, A237591, A237593, A244050, A245092, A262626, A279387, A279388, A279391.
%K A281010 sign,tabl
%O A281010 1,61
%A A281010 _Omar E. Pol_, Jan 12 2017
cp's OEIS Frontend

A281010 Triangle read by rows in which row 2n-1 lists the widths of the symmetric representation of sigma(n), and row 2n lists a finite sequence S together with -1, with the property that the partial sums of S give the row 2n-1.
