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

1,2

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

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(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)

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

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).

Sequence in context: A120246 A361261 A120244 * A341511 A089308 A305579

Adjacent sequences: A094724 A094725 A094726 * A094728 A094729 A094730

Keyword

nonn,tabl,easy

Author

Reinhard Zumkeller, May 24 2004

Status

approved