A376133 Triangle T read by rows: T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 and T(n, k) = T(n, k-1) + (-1)^k * 2 * (n+1-k) for k >= 2.

1, 2, 4, 3, 7, 5, 6, 12, 8, 10, 9, 17, 11, 15, 13, 14, 24, 16, 22, 18, 20, 19, 31, 21, 29, 23, 27, 25, 26, 40, 28, 38, 30, 36, 32, 34, 33, 49, 35, 47, 37, 45, 39, 43, 41, 42, 60, 44, 58, 46, 56, 48, 54, 50, 52, 51, 71, 53, 69, 55, 67, 57, 65, 59, 63, 61, 62, 84, 64, 82, 66, 80, 68, 78, 70, 76, 72, 74

Offset: 1

Author

Werner Schulte, Sep 11 2024

Keywords

• nonn
• easy
• tabl

Comments

Row n consists of the next n odd/even natural numbers if n is odd/even. So the sequence yields a permutation of the natural numbers.

Examples

			Row n=5: Next (1,3,5,7 see rows 1 and 3) five odd numbers are 9,11,13,15 and 17; with "9+8-6+4-2" we get 9,17,11,15,13 for row 5.
Row n=8: Next (2,4,..,24 see rows 2, 4 and 6) eight even numbers are 26,28,..,40; with "26+14-12+10-8+6-4+2" we get 26,40,28,38,30,36,32,34 for row 8.
Triangle T(n, k) for 1 <= k <= n starts:
n\ k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   2   4
   3 :   3   7   5
   4 :   6  12   8  10
   5 :   9  17  11  15  13
   6 :  14  24  16  22  18  20
   7 :  19  31  21  29  23  27  25
   8 :  26  40  28  38  30  36  32  34
   9 :  33  49  35  47  37  45  39  43  41
  10 :  42  60  44  58  46  56  48  54  50  52
  11 :  51  71  53  69  55  67  57  65  59  63  61
  12 :  62  84  64  82  66  80  68  78  70  76  72  74
  etc.
		

Crossrefs

Cf. A061925 (column 1), A074148 (column 2), A074149 (row sums), A236283 (main diagonal).

Programs

Maple

T := (n, k) -> ((-1)^k*(2 + 4*(n - k)) + 2*n^2 + (-1)^n + 5)/4:
seq(seq(T(n, k), k = 1..n), n = 1..12);  # Peter Luschny, Sep 13 2024

PARI

T(n,k)=(2*n*n+(-1)^k*4*(n-k)+5+2*(-1)^k+(-1)^n)/4

Formula

T(n, k) = (2*n*n + (-1)^k * 4 * (n - k) + 5 + 2 * (-1)^k + (-1)^n) / 4.

T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 = A061925(n-1).

T(n, 2) = (2*n*n + 4*n - 1 + (-1)^n) / 4 = A074148(n) for n > 1.

T(n, k) = T(n, k-2) - (-1)^k * 2 for 3 <= k <= n.

G.f.: x*y*(1 + 2*x*y + 2*x^5*y^2 + x^6*y^3 - x^4*y*(3 + y + y^2) - x^2*(1 + y + 3*y^2) + 2*x^3*(1 + y^3))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Sep 12 2024