A377802 Triangle read by rows: T(n, k) = (2 * (n+1)^2 + 7 - (-1)^n) / 8 - k.

1, 2, 1, 4, 3, 2, 6, 5, 4, 3, 9, 8, 7, 6, 5, 12, 11, 10, 9, 8, 7, 16, 15, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31

Offset: 1

Author

Werner Schulte, Nov 07 2024

Keywords

• nonn
• easy
• tabl

Comments

The natural numbers, based on quarter-squares (A002620 and A033638); every natural number occurs exactly twice.

Examples

			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 :   2   1
   3 :   4   3   2
   4 :   6   5   4   3
   5 :   9   8   7   6   5
   6 :  12  11  10   9   8   7
   7 :  16  15  14  13  12  11  10
   8 :  20  19  18  17  16  15  14  13
   9 :  25  24  23  22  21  20  19  18  17
  10 :  30  29  28  27  26  25  24  23  22  21
  11 :  36  35  34  33  32  31  30  29  28  27  26
  12 :  42  41  40  39  38  37  36  35  34  33  32  31
  13 :  49  48  47  46  45  44  43  42  41  40  39  38  37
  etc.
		

Crossrefs

A002620 (column 1), A024206 (column 2), A014616 (column 3), A004116 (column 4), A033638 (main diagonal), A290743 (1st subdiagonal).

Cf. A006003, A026741, A002061, A000290, A002378, A005563, A246694.

Programs

PARI

T(n,k)=(2*(n+1)^2+7-(-1)^n)/8-k

Formula

T(n, k) = A002620(n+1) + 1 - k.

T(2*n-1, n) = n^2 - n + 1 = A002061(n); T(2*n-2, n) = (n-1)^2 = A000290(n-1) for n > 1; T(2*n-3, n) = (n-1) * (n-2) = A002378(n-2) for n > 2; T(2*n-4, n) = (n-1) * (n-3) = A005563(n-3) for n > 3.

Row sums are (2 * n^3 + 5 * n - n * (-1)^n) / 8 = (A006003(n) + A026741(n)) / 2.

G.f.: x*y*(1 - x*y + x^2*y + x^4*y^2 - x^5*y^3 + x^6*y^3 - x^3*y*(1 + 2*y - y^2))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Nov 08 2024