A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

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, 23

Offset: 1

Author

Reinhard Zumkeller, May 24 2004

Keywords

• nonn
• tabl
• easy

Comments

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

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

Examples

			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
		

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.

Crossrefs

Cf. A002024, A002260, A003056, A004526, A004736, A005408, A005843.

Cf. A016777, A016813, A016861, A037213, A076100, A078112, A093005.

Cf. A094728, A214604, A214661, A319556.

Cf. A128076 (rows reversed).

Programs

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

Magma

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

Mathematica

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

Python

from math import isqrt
def A094727(n): return ((a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-a)>>1)+n-1 # Chai Wah Wu, Jun 19 2025

SageMath

flatten([[n+k for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

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(n), n) = A016777(n), n >= 0.

Sum_{k=1..n} T(n, k) = A000326(n) (row sums).

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

T(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)

From G. C. Greubel, Mar 10 2024: (Start)

T(3*n-3, n) = A016813(n-1).

T(4*n-4, n) = A016861(n-1).

Sum_{k=0..n-1} (-1)^k*T(n, k) = A319556(n).

Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A093005(n).

Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = A078112(n-1).

Sum_{j=1..n} (Sum_{k=0..n-1} T(j, k)) = A002411(n) (sum of n rows). (End)