A349890 Triangle read by rows: T(n,k) = n * 2^e(n) - (4^e(n) - 1) / 3 - k * (k - 1) / 2 with e(n) = 1 + floor(log_2(n)) for n >= 1 and 1 <= k <= n.

1, 3, 2, 7, 6, 4, 11, 10, 8, 5, 19, 18, 16, 13, 9, 27, 26, 24, 21, 17, 12, 35, 34, 32, 29, 25, 20, 14, 43, 42, 40, 37, 33, 28, 22, 15, 59, 58, 56, 53, 49, 44, 38, 31, 23, 75, 74, 72, 69, 65, 60, 54, 47, 39, 30, 91, 90, 88, 85, 81, 76, 70, 63, 55, 46, 36, 107, 106, 104, 101, 97, 92, 86, 79, 71, 62, 52, 41

Offset: 1

Author

Werner Schulte, Dec 04 2021

Keywords

• nonn
• easy
• tabl

Comments

Conjecture: The terms of the triangle yield a permutation of the positive integers (A000027).

Examples

			The triangle T(n, k) for 1 <= k <= n begins:
n\k:   1   2   3   4   5   6   7   8   9  10  11
================================================
01 :   1
02 :   3   2
03 :   7   6   4
04 :  11  10   8   5
05 :  19  18  16  13   9
06 :  27  26  24  21  17  12
07 :  35  34  32  29  25  20  14
08 :  43  42  40  37  33  28  22  15
09 :  59  58  56  53  49  44  38  31  23
10 :  75  74  72  69  65  60  54  47  39  30
11 :  91  90  88  85  81  76  70  63  55  46  36
etc.
		

Crossrefs

Cf. A000027, A000217, A007583.

Programs

PARI

T(n,k) = my(e=1+logint(n,2)); n*2^e - (4^e-1)/3 - k*(k-1)/2;
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Dec 05 2021

Formula

T(2^n, 1) = A007583(n) for n >= 0.

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

T(n, k) = T(n-1, k) + T(n-1, k-1) - T(n-1-2^(e(n-1)-e(n-2)), k-1) with e(n) = 1 + floor(log_2(n)) for n > 3 and 1 < k < n-1 (conjectured).