

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
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OFFSET

1,2


COMMENTS

row sums give A000326;
all numbers m occur ceiling(m/2) times, see A004526.
The LCM of the nth 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, nA005843(k)) = A005843(nk) for 0 <= k <= n/2;
T(n, nA005408(k)) = A005408(nk) 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(8n7)+1)/2), a(n) = n  m(m3)/2  1.  Carl R. White, Jul 30 2009
T(m,n) = m+n1, 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=nt*(t+1)/2, t=floor((1+sqrt(8*n7))/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+n1: m in [1..n] ]: n in [1..z] ];
(Haskell)
a094727 n k = n + k
a094727_row n = a094727_tabl !! (n1)
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



