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%I #27 Dec 01 2020 16:17:27
%S 1,3,5,1,7,1,9,3,11,3,1,13,5,1,15,5,1,17,7,3,19,7,3,1,21,9,3,1,23,9,5,
%T 1,25,11,5,1,27,11,5,3,29,13,7,3,1,31,13,7,3,1,33,15,7,3,1,35,15,9,5,
%U 1,37,17,9,5,1,39,17,9,5,3,41,19,11,5,3,1,43,19,11,7,3,1
%N Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists the odd numbers k times, and the first element of column k is in row k(k+1)/2.
%C A missing companion to A196020 and A235791.
%C T(n,k) is the total number of horizontal steps in the first n levels of the k-th largest double-staircase of the diagram defined in A335616 (see example). - _Omar E. Pol_, Nov 30 2020
%C Column k is the partial sums of the k-th column of A339275. - _Omar E. Pol_, Dec 01 2020
%H Alois P. Heinz, <a href="/A338721/b338721.txt">Rows n = 1..500, flattened</a>
%F T(n,k) = 2 * floor((n-k*(k-1)/2)/k) - 1. - _Alois P. Heinz_, Nov 30 2020
%e Triangle begins:
%e 1;
%e 3;
%e 5, 1;
%e 7, 1;
%e 9, 3;
%e 11, 3, 1;
%e 13, 5, 1;
%e 15, 5, 1;
%e 17, 7, 3;
%e 19, 7, 3, 1;
%e 21, 9, 3, 1;
%e 23, 9, 5, 1;
%e 25, 11, 5, 1;
%e 27, 11, 5, 3;
%e 29, 13, 7, 3, 1;
%e 31, 13, 7, 3, 1;
%e 33, 15, 7, 3, 1;
%e 35, 15, 9, 5, 1;
%e 37, 17, 9, 5, 1;
%e 39, 17, 9, 5, 3;
%e 41, 19, 11, 5, 3, 1;
%e 43, 19, 11, 7, 3, 1;
%e 45, 21, 11, 7, 3, 1;
%e 47, 21, 13, 7, 3, 1;
%e 49, 23, 13, 7, 5, 1;
%e 51, 23, 13, 9, 5, 1;
%e 53, 25, 15, 9, 5, 3;
%e 55, 25, 15, 9, 5, 3, 1;
%e ...
%e From _Omar E. Pol_, Nov 30 2020: (Start)
%e For an illustration of the rows of triangle consider the infinite "double-staircases" diagram defined in A335616.
%e For n = 15 the diagram with first 15 levels looks like this:
%e .
%e Level "Double-staircases" diagram
%e . _
%e 1 _|1|_
%e 2 _|1 _ 1|_
%e 3 _|1 |1| 1|_
%e 4 _|1 _| |_ 1|_
%e 5 _|1 |1 _ 1| 1|_
%e 6 _|1 _| |1| |_ 1|_
%e 7 _|1 |1 | | 1| 1|_
%e 8 _|1 _| _| |_ |_ 1|_
%e 9 _|1 |1 |1 _ 1| 1| 1|_
%e 10 _|1 _| | |1| | |_ 1|_
%e 11 _|1 |1 _| | | |_ 1| 1|_
%e 12 _|1 _| |1 | | 1| |_ 1|_
%e 13 _|1 |1 | _| |_ | 1| 1|_
%e 14 _|1 _| _| |1 _ 1| |_ |_ 1|_
%e 15 |1 |1 |1 | |1| | 1| 1| 1|
%e .
%e The first largest double-staircase has 29 horizontal steps, the second double-staircase has 13 steps, the third double-staircase has 7 steps, the fourth double-staircase has 3 steps and the fifth double-staircase has only one step, so the 15th row of triangle is [29, 13, 7, 3, 1]. (End)
%p T:= (n, k)-> 2*iquo(n-k*(k-1)/2, k)-1:
%p seq(seq(T(n,k), k=1..floor((sqrt(1+8*n)-1)/2)), n=1..30); # _Alois P. Heinz_, Nov 30 2020
%Y Cf. A196020, A235791, A237593, A335616, A338722, A338723, A339275.
%K nonn,tabf
%O 1,2
%A _N. J. A. Sloane_, Nov 30 2020.