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A361819 Irregular triangle read by rows where T(n,k) is the distance which number A361660(n,k) moves in the process described in A361642.

%I A361819 #24 Apr 13 2023 08:30:37
%S A361819 2,3,3,4,2,2,4,5,3,4,3,5,6,4,2,3,3,2,4,6,7,5,3,5,2,5,3,5,7,8,6,4,2,4,
%T A361819 4,4,4,2,4,6,8,9,7,5,3,6,3,3,3,3,6,3,5,7,9,10,8,6,4,2,5,5,2,6,2,5,5,2,
%U A361819 4,6,8,10,11,9,7,5,3,7,4,4,5,5,4,4,7,3,5,7,9,11
%N A361819 Irregular triangle read by rows where T(n,k) is the distance which number A361660(n,k) moves in the process described in A361642.
%C A361819 Number A361660(n,k) moves to the right and then down and T(n,k) counts the steps in both.
%C A361819 All moves are T(n,k) >= 2 steps since a number moves at least one step right and one step down.
%C A361819 Row n has sum A002378(n-1) which is the total steps to move a column down to a row irrespective of the order of movement.
%C A361819 Each row is a palindrome (the same when reversed), since the moves in A361642 are exactly the reverse moves to send its row back to the starting column.
%e A361819 Irregular triangle T(n,k) begins:
%e A361819   n/k     |   1    2    3    4    5    6    7    8    9
%e A361819   ------------------------------------------------------
%e A361819   1       |   (empty row)
%e A361819   2       |   2;
%e A361819   3       |   3,   3;
%e A361819   4       |   4,   2,   2,   4;
%e A361819   5       |   5,   3,   4,   3,   5;
%e A361819   6       |   6,   4,   2,   3,   3,   2,   4,   6;
%e A361819   7       |   7,   5,   3,   5,   2,   5,   3,   5,   7;
%e A361819  ...
%o A361819 (MATLAB)
%o A361819 function a = A361819( max_row )
%o A361819     k = 1;
%o A361819     for r = 2:max_row
%o A361819         h = zeros(1,r); h(1) = r;
%o A361819         while max(h) > 1
%o A361819            j =  find(h == max(h), 1, 'last' );
%o A361819            m =  find(h < max(h)-1, 1, 'first' );
%o A361819            a(k) = (m-j) + (h(j)-h(m)) - 1;
%o A361819            h(j) = h(j) - 1; h(m) = h(m) + 1;
%o A361819            k = k+1;
%o A361819         end
%o A361819     end
%o A361819 end % _Thomas Scheuerle_, Mar 27 2023
%Y A361819 Cf. A361642, A361660, A002541 (row lengths), A002378 (row sums).
%K A361819 nonn,tabf
%O A361819 1,1
%A A361819 _Tamas Sandor Nagy_, Mar 25 2023