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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 (list; table; graph; refs; listen; history; text; internal format)
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.

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 A268680 A126974

Adjacent sequences:  A108869 A108870 A108871 * A108873 A108874 A108875

KEYWORD

easy,nonn,tabl

AUTHOR

Andrew Plewe, Jul 13 2005

EXTENSIONS

Offset changed by Reinhard Zumkeller, Oct 01 2014

STATUS

approved

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Last modified August 20 06:32 EDT 2017. Contains 290824 sequences.