A108872 Sums of ordinal references for a triangular table read by columns, top to bottom.

2, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 16, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22

Offset: 1

Author

Andrew S. Plewe, Jul 13 2005

Keywords

• easy
• nonn
• tabl

Comments

The ordinal references (i,j) for a triangular table are arranged as follows:

(1,1) (2,1) (3,1)

..... (2,2) (3,2)

........... (3,3)

The sequence comprises the sum of each reference in each column, read top to bottom. A similar sequence is A003057, which consists of the sums of the ordinal references for an array read by antidiagonals.

Subtriangle of triangle in A051162. - Philippe Deléham, Mar 26 2013

First 9 rows coincide with triangle A248110; T(n,k) = A002260(n,k) + n; T(2*n-1,n) = A016789(n-1). - Reinhard Zumkeller, Oct 01 2014

Examples

			a(1) = (1,1) = 1 + 1 = 2
a(2) = (2,1) = 2 + 1 = 3
a(3) = (2,2) = 2 + 2 = 4
a(4) = (3,1) = 3 + 1 = 4, etc.
Triangle begins:
  2
  3, 4
  4, 5, 6
  5, 6, 7, 8
  6, 7, 8, 9, 10
  7, 8, 9, 10, 11, 12
  8, 9, 10, 11, 12, 13, 14
  9, 10, 11, 12, 13, 14, 15, 16
  ... - _Philippe Deléham_, Mar 26 2013
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
• NRICH discussion thread, Floor Function Identity. [Via Wayback Machine]

Crossrefs

Cf. A003057.

Cf. A002024, A002260.

Cf. A005408, A005843.

Cf. A016789 (central terms), A248110.

Programs

Haskell

a108872 n k = a108872_tabl !! (n-1) !! (k-1)
a108872_row n = a108872_tabl !! (n-1)
a108872_tabl = map (\x -> [x + 1 .. 2 * x]) [1..]
-- Reinhard Zumkeller, Oct 01 2014

Mathematica

Flatten[ Table[i + j, {j, 1, 12}, {i, 1, j}]] (* Jean-François Alcover, Oct 07 2011 *)

Python

from math import isqrt
def A108872(n): return n+((r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-r)>>1) # Chai Wah Wu, Nov 08 2024

Formula

a(n) = a(i, j) = i + j

a(n) = A002024(n) + A002260(n) = floor(1/2 + sqrt(2n)) + n - (m(m+1)/2) + 1, where m = floor((sqrt(8n+1) - 1) / 2 ). The floor function may be computed directly by using the expression floor(x) = x + (arctan(cot(Pi*x)) / Pi) - 1/2 (equation from nrich.maths.org, see links).

Sum_{k=0..n} T(n,k) = A005449(n+1). - Philippe Deléham, Mar 26 2013

Extensions

Offset changed by Reinhard Zumkeller, Oct 01 2014