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 A280850 #168 Jun 19 2019 17:56:06
%S A280850 1,3,2,2,7,0,3,3,11,0,1,4,4,0,15,0,0,5,5,3,9,0,0,9,6,6,0,0,23,0,5,0,7,
%T A280850 7,0,0,12,0,0,12,8,8,7,0,1,31,0,0,0,0,9,9,0,0,0,35,0,2,2,0,10,10,0,0,
%U A280850 0,39,0,0,0,3,11,11,5,0,0,5,18,0,0,18,0,0,12,12,0,0,0,0,47,0,13,0,0,0
%N A280850 Irregular triangle read by rows in which row n is constructed with an algorithm using the n-th row of triangle A196020 (see Comments for precise definition).
%C A280850 For the construction of the n-th row of this triangle start with a copy of the n-th row of the triangle A196020.
%C A280850 Then replace each element of the m-th pair of positive integers (x, y) with the value (x - y)/2, where "y" is the m-th even-indexed term of the row, and "x" is its previous nearest odd-indexed term not used in another pair in the same row, if such a pair exist. Otherwise T(n,k) = A196020(n,k). (See example).
%C A280850 Observation 1: at least for the first 28 rows of the triangle the nonzero terms in the n-th row are also the subparts of the symmetric representation of sigma(n), assuming the ordering of the subparts in the same row does not matter.
%C A280850 Question 1: are always the nonzero terms of the n-th row the same as all the subparts of the symmetric representation of sigma(n)? If not, what is the index of the row in which appears the first counterexample?
%C A280850 Note that the "subparts" are the regions that arise after the dissection of the symmetric representation of sigma(n) into successive layers of width 1.
%C A280850 For more information about "subparts" see A279387 and A237593.
%C A280850 About the question 1, it appears that the n-th row of the triangle A280851 and the n-th row of this triangle contain the same nonzero numbers, though in different order; checked through n = 250000. - _Hartmut F. W. Hoft_, Jan 31 2018
%C A280850 From _Omar E. Pol_, Feb 02 2018: (Start)
%C A280850 Observation 2: at least for the first 28 rows of the triangle we have that in the n-th row the odd-indexed terms, from left to right, together with the even-indexed terms, from right to left, form a finite sequence in which the nonzero terms are the same as the n-th row of triangle A280851, which lists the subparts of the symmetric representation of sigma(n).
%C A280850 Question 2: Are always the same for all rows? If not, what is the index of the row in which appears the first counterexample? (End)
%C A280850 Conjecture: the odd-indexed terms of the n-th row together with the even-indexed terms of the same row but listed in reverse order give the n-th row of triangle A296508 (this is the same conjecture from A296508). - _Omar E. Pol_, Apr 20 2018
%H A280850 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A280850 Triangle begins (rows 1..28):
%e A280850 1;
%e A280850 3;
%e A280850 2, 2;
%e A280850 7, 0;
%e A280850 3, 3;
%e A280850 11, 0, 1;
%e A280850 4, 4, 0;
%e A280850 15, 0, 0;
%e A280850 5, 5, 3;
%e A280850 9, 0, 0, 9;
%e A280850 6, 6, 0, 0;
%e A280850 23, 0, 5, 0;
%e A280850 7, 7, 0, 0;
%e A280850 12, 0, 0, 12;
%e A280850 8, 8, 7, 0, 1;
%e A280850 31, 0, 0, 0, 0;
%e A280850 9, 9, 0, 0, 0;
%e A280850 35, 0, 2, 2, 0;
%e A280850 10, 10, 0, 0, 0;
%e A280850 39, 0, 0, 0, 3;
%e A280850 11, 11, 5, 0, 0, 5;
%e A280850 18, 0, 0, 18, 0, 0;
%e A280850 12, 12, 0, 0, 0, 0;
%e A280850 47, 0, 13, 0, 0, 0;
%e A280850 13, 13, 0, 0, 5, 0;
%e A280850 21, 0, 0, 21, 0, 0;
%e A280850 14, 14, 6, 0, 0, 6;
%e A280850 55, 0, 0, 0, 0, 0, 1;
%e A280850 ...
%e A280850 An example of the algorithm.
%e A280850 For n = 75, the construction of the 75th row of this triangle is as shown below:
%e A280850 .
%e A280850 75th row of A196020: [149, 73, 47, 0, 25, 19, 0, 0, 0, 5, 0]
%e A280850 .
%e A280850 Odd-indexed terms: 149 47 25 0 0 0
%e A280850 Even-indexed terms: 73 0 19 0 5
%e A280850 .
%e A280850 First even-indexed nonzero term: 73
%e A280850 First pair: 149 73
%e A280850 . *----*
%e A280850 Difference: 149 - 73 = 76
%e A280850 76/2 = 38 *----*
%e A280850 New first pair: 38 38
%e A280850 .
%e A280850 Second even-indexed nonzero term: 19
%e A280850 Second pair: 25 19
%e A280850 . *---*
%e A280850 Difference: 25 - 19 = 6
%e A280850 6/2 = 3 *---*
%e A280850 New second pair: 3 3
%e A280850 .
%e A280850 Third even-indexed nonzero term: 5
%e A280850 Third pair: 47 5
%e A280850 . *----------------------*
%e A280850 Difference: 47 - 5 = 42
%e A280850 42/2 = 21 *----------------------*
%e A280850 New third pair: 21 21
%e A280850 .
%e A280850 So the 75th row
%e A280850 of this triangle is [38, 38, 21, 0, 3, 3, 0, 0, 0, 21, 0]
%e A280850 .
%e A280850 On the other hand, the 75th row of A237593 is [38, 13, 7, 4, 3, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 3, 4, 7, 13, 38], and the 74th row of the same triangle is [38, 13, 6, 5, 3, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 3, 5, 6, 13, 38], therefore between both symmetric Dyck paths (described in A237593 and A279387) there are six subparts: [38, 38, 21, 21, 3, 3]. (The diagram of the symmetric representation of sigma(75) is too large to include.) At least in this case the nonzero terms of the 75th row of the triangle coincide with the subparts of the symmetric representation of sigma(75). The ordering of the elements does not matter.
%e A280850 Continuing with the original example, in the 75th row of this triangle we have that the odd-indexed terms, from left to right, together with the even-indexed terms, from right to left, form the finite sequence [38, 21, 3, 0, 0, 0, 21, 0, 3, 0, 38] which is the 75th row of a triangle. At least in this case the nonzero terms coincide with the 75th row of triangle A280851: [38, 21, 3, 21, 3, 38], which lists the six subparts of the symmetric representation of sigma(75) in order of appearance from left to right. - _Omar E. Pol_, Feb 02 2018
%e A280850 In accordance with the conjecture from the Comments section, the finite sequence [38, 21, 3, 0, 0, 0, 21, 0, 3, 0, 38] mentioned above should be the 75th row of triangle A296508. - _Omar E. Pol_, Apr 20 2018
%t A280850 (* functions row[], line[] and their support are defined in A196020 *)
%t A280850 (* maintain a stack of odd indices with nonzero entries for matching *)
%t A280850 a280850[n_] := Module[{a=line[n], r=row[n], stack={1}, i, j, b}, For[i=2, i<=r, i++, If[a[[i]]!=0, If[OddQ[i], AppendTo[stack, i], j=Last[stack]; b=(a[[j]]-a[[i]])/2; a[[i]]=b; a[[j]]=b; stack=Drop[stack, -1]]]]; a]
%t A280850 Flatten[Map[a280850,Range[24]]] (* data *)
%t A280850 TableForm[Map[a280850, Range[28]], TableDepth->2] (* triangle in Example *)
%t A280850 (* _Hartmut F. W. Hoft_, Jan 31 2018 *)
%Y A280850 Row sums give A000203.
%Y A280850 The number of positive terms in row n is A001227(n).
%Y A280850 Row n has length A003056(n).
%Y A280850 Column k starts in row A000217(k).
%Y A280850 Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A239660, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280851, A296508.
%K A280850 nonn,tabf
%O A280850 1,2
%A A280850 _Omar E. Pol_, Jan 09 2017
%E A280850 Name edited by _Omar E. Pol_, Nov 11 2018