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

1,2

Links

Table of n, a(n) for n=1..78.

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

Example

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.

Prog

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

Crossrefs

Cf. A000217, A006003, A176040.

Sequence in context: A131005 A122361 A131162 * A131172 A057503 A074684

Adjacent sequences: A364387 A364388 A364389 * A364391 A364392 A364393

Keyword

nonn,easy,tabl

Author

Werner Schulte, Jul 21 2023

Status

approved