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

1,1

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

Crossrefs

Cf. A003057.

Cf. A002024, A002260.

Cf. A005408, A005843.

Cf. A016789 (central terms), A248110.

Sequence in context: A101504 A125568 A248110 * A147847 A370386 A335572

Adjacent sequences: A108869 A108870 A108871 * A108873 A108874 A108875

Keyword

easy,nonn,tabl

Author

Andrew S. Plewe, Jul 13 2005

Extensions

Offset changed by Reinhard Zumkeller, Oct 01 2014

Status

approved