login
A364390
Triangle T(n, k) based on A176040 which read by rows yields a permutation of the positive integers.
1
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
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
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, Jul 21 2023
STATUS
approved