

A108872


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


4



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
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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*n1,n) = A016789(n1).  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}]] (* JeanFrançois Alcover, Oct 07 2011 *)


PROG

(Haskell)
a108872 n k = a108872_tabl !! (n1) !! (k1)
a108872_row n = a108872_tabl !! (n1)
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 A335572 A268680
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



