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A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n. 21
1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

row sums give A000326;

all numbers m occur ceiling(m/2) times, see A004526.

The LCM of the n-th row is A076100. - Michel Marcus, Mar 18 2018

LINKS

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Bruno Berselli, Illustration of the initial terms.

László Németh, On the Binomial Interpolated Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.8.

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

FORMULA

T(n+1,k) = T(n,k)+1 = T(n,k+1); T(n+1,k+1) = T(n,k)+2;

T(n, n-A005843(k)) = A005843(n-k) for 0 <= k <= n/2;

T(n, n-A005408(k)) = A005408(n-k) for 0 <= k < n/2;

T(A005408(k),k) = A016777(k);

T(n,k) = A002024(n,k) + A002260(n,k) - 1. - Reinhard Zumkeller, Apr 27 2006

As a sequence rather than as a table: If m = floor((sqrt(8n-7)+1)/2), a(n) = n - m(m-3)/2 - 1. - Carl R. White, Jul 30 2009

T(m,n) = m+n-1, m >= n >= 1. - Vincenzo Librandi, Nov 23 2009 [corrected by Klaus Brockhaus, Nov 23 2009]

a(n,k) = A037213((A214604(n,k) + A214661(n,k)) / 2). - Reinhard Zumkeller, Jul 25 2012

From Boris Putievskiy, Jan 16 2013: (Start)

a(n) = A002260(n) + A003056(n).

a(n) = i+t, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)

EXAMPLE

Triangle begins:

1;

2, 3;

3, 4, 5;

4, 5, 6, 7;

5, 6, 7, 8, 9;

6, 7, 8, 9, 10, 11;

7, 8, 9, 10, 11, 12, 13;

8, 9, 10, 11, 12, 13, 14, 15;

9, 10, 11, 12, 13, 14, 15, 16, 17;

... - Philippe Deléham, Mar 30 2013

MATHEMATICA

Table[n + Range[0, n - 1], {n, 12}] // Flatten (* Michael De Vlieger, Dec 16 2016 *)

PROG

(Magma) z:=12; &cat[ [m+n-1: m in [1..n] ]: n in [1..z] ];

(Haskell)

a094727 n k = n + k

a094727_row n = a094727_tabl !! (n-1)

a094727_tabl = iterate (\row@(h:_) -> (h + 1) : map (+ 2) row) [1]

-- Reinhard Zumkeller, Jul 22 2012

CROSSREFS

Cf. A002260, A004736, A094728, A128076 (rows reversed).

Sequence in context: A120245 A120246 A120244 * A341511 A089308 A305579

Adjacent sequences: A094724 A094725 A094726 * A094728 A094729 A094730

KEYWORD

nonn,tabl,easy

AUTHOR

Reinhard Zumkeller, May 24 2004

STATUS

approved

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Last modified December 7 03:44 EST 2022. Contains 358649 sequences. (Running on oeis4.)