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A364390 Triangle T(n, k) based on A176040 which read by rows yields a permutation of the positive integers.

%I A364390 #12 Dec 04 2023 06:32:10
%S A364390 1,3,2,8,7,4,10,9,6,5,19,18,15,14,11,21,20,17,16,13,12,34,33,30,29,26,
%T A364390 25,22,36,35,32,31,28,27,24,23,53,52,49,48,45,44,41,40,37,55,54,51,50,
%U A364390 47,46,43,42,39,38,76,75,72,71,68,67,64,63,60,59,56,78,77,74,73,70,69,66,65,62,61,58,57
%N A364390 Triangle T(n, k) based on A176040 which read by rows yields a permutation of the positive integers.
%F A364390 T(n, k) = n*(n+1)/2 + (n-1)*(n mod 2) - 2*k + 3 - (k mod 2) for 1 <= k <= n.
%F A364390 T(n, 1) = n*(n+1)/2 + (n-1)*(n mod 2) for n > 0.
%F A364390 T(2*n, 1) = A000217(2*n) for n > 0.
%F A364390 T(n, k) - T(n, k+1) = A176040(k) for k > 0.
%F A364390 T(n, k) = T(n-1, k) + T(n, k-1) - T(n-1, k-1) for 1 < k < n.
%F A364390 T(2*n, k) - T(2*n-1, k) = 2 for 1 <= k < 2*n.
%F A364390 Row sums: A006003(n) - (-1)^n * 2 * floor((n-1)/2) * (1 + floor((n-1)/2)) for n > 0. - _Werner Schulte_, Dec 03 2023
%e A364390 Triangle T(n, k) for 1 <= k <= n begins:
%e A364390 n\k:    1    2    3    4    5    6    7    8    9   10   11   12   13   14
%e A364390 ==========================================================================
%e A364390 01 :    1
%e A364390 02 :    3    2
%e A364390 03 :    8    7    4
%e A364390 04 :   10    9    6    5
%e A364390 05 :   19   18   15   14   11
%e A364390 06 :   21   20   17   16   13   12
%e A364390 07 :   34   33   30   29   26   25   22
%e A364390 08 :   36   35   32   31   28   27   24   23
%e A364390 09 :   53   52   49   48   45   44   41   40   37
%e A364390 10 :   55   54   51   50   47   46   43   42   39   38
%e A364390 11 :   76   75   72   71   68   67   64   63   60   59   56
%e A364390 12 :   78   77   74   73   70   69   66   65   62   61   58   57
%e A364390 13 :  103  102   99   98   95   94   91   90   87   86   83   82   79
%e A364390 14 :  105  104  101  100   97   96   93   92   89   88   85   84   81   80
%e A364390 etc.
%o A364390 (PARI) T(n,k) = n*(n+1)/2 + (n-1)*(n%2) - 2*k + 3 - (k%2)
%Y A364390 Cf. A000217, A006003, A176040.
%K A364390 nonn,easy,tabl
%O A364390 1,2
%A A364390 _Werner Schulte_, Jul 21 2023