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

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, 25, 22, 36, 35, 32, 31, 28, 27, 24, 23, 53, 52, 49, 48, 45, 44, 41, 40, 37, 55, 54, 51, 50, 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

Offset: 1

Author

Werner Schulte, Jul 21 2023

Keywords

• nonn
• easy
• tabl

Examples

			Triangle T(n, k) for 1 <= k <= n begins:
n\k:    1    2    3    4    5    6    7    8    9   10   11   12   13   14
==========================================================================
01 :    1
02 :    3    2
03 :    8    7    4
04 :   10    9    6    5
05 :   19   18   15   14   11
06 :   21   20   17   16   13   12
07 :   34   33   30   29   26   25   22
08 :   36   35   32   31   28   27   24   23
09 :   53   52   49   48   45   44   41   40   37
10 :   55   54   51   50   47   46   43   42   39   38
11 :   76   75   72   71   68   67   64   63   60   59   56
12 :   78   77   74   73   70   69   66   65   62   61   58   57
13 :  103  102   99   98   95   94   91   90   87   86   83   82   79
14 :  105  104  101  100   97   96   93   92   89   88   85   84   81   80
etc.
		

Crossrefs

Cf. A000217, A006003, A176040.

Programs

PARI

T(n,k) = n*(n+1)/2 + (n-1)*(n%2) - 2*k + 3 - (k%2)

Formula

T(n, k) = n*(n+1)/2 + (n-1)*(n mod 2) - 2*k + 3 - (k mod 2) for 1 <= k <= n.

T(n, 1) = n*(n+1)/2 + (n-1)*(n mod 2) for n > 0.

T(2*n, 1) = A000217(2*n) for n > 0.

T(n, k) - T(n, k+1) = A176040(k) for k > 0.

T(n, k) = T(n-1, k) + T(n, k-1) - T(n-1, k-1) for 1 < k < n.

T(2*n, k) - T(2*n-1, k) = 2 for 1 <= k < 2*n.

Row sums: A006003(n) - (-1)^n * 2 * floor((n-1)/2) * (1 + floor((n-1)/2)) for n > 0. - Werner Schulte, Dec 03 2023