OFFSET
1,1
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
LINKS
Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
NRICH discussion thread, Floor Function Identity. [Via Wayback Machine]
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
EXAMPLE
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
MATHEMATICA
Flatten[ Table[i + j, {j, 1, 12}, {i, 1, j}]] (* Jean-François Alcover, Oct 07 2011 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Andrew S. Plewe, Jul 13 2005
EXTENSIONS
Offset changed by Reinhard Zumkeller, Oct 01 2014
STATUS
approved