A385406 Triangle read by rows: T(n, k) = n*(n+1)/2 - floor((n-1)/2) - (-1)^k * floor(k/2).

1, 3, 2, 5, 4, 6, 9, 8, 10, 7, 13, 12, 14, 11, 15, 19, 18, 20, 17, 21, 16, 25, 24, 26, 23, 27, 22, 28, 33, 32, 34, 31, 35, 30, 36, 29, 41, 40, 42, 39, 43, 38, 44, 37, 45, 51, 50, 52, 49, 53, 48, 54, 47, 55, 46, 61, 60, 62, 59, 63, 58, 64, 57, 65, 56, 66, 73, 72, 74, 71, 75, 70, 76, 69, 77, 68, 78, 67

Offset

1,2

Comments

This triangle seen as a sequence yields a permutation of the natural numbers (A000027).

Links

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

Index entries for sequences that are permutations of the natural numbers

Formula

T(n, k) = T(n, k-1) - (-1)^k * (k-1) for 1 < k <= n with initial values T(n, 1) = n*(n+1)/2 - floor((n-1)/2) for n >= 1.

T(n, n) = n*(n+1)/2 + (1-n) * (1 - n mod 2) = A128918(n).

T(2*n-1, n) = 2*n^2 - 2*n + 1 - (-1)^n * floor(n/2) = A213399(n-1).

Example

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

Mathematica

T[n_, k_] := n*(n+1)/2 - Floor[(n-1)/2] - (-1)^k*Floor[k/2]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 28 2025 *)

Prog

(PARI) T(n, k) = n*(n+1)/2 - floor((n-1)/2) - (-1)^k * floor(k/2)

Crossrefs

Cf. A080827 (column 1), A128918 (main diagonal), A006003 (row sums), A213399.

Sequence in context: A194836 A054069 A194869 * A191736 A372654 A297208

Adjacent sequences: A385403 A385404 A385405 * A385407 A385408 A385409

Keyword

nonn,easy,tabl

Author

Werner Schulte, Jun 27 2025

Status

approved