A127739 Triangle read by rows, in which row n contains the triangular number T(n) = A000217(n) repeated n times; smallest triangular number greater than or equal to n.

1, 3, 3, 6, 6, 6, 10, 10, 10, 10, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 28, 28, 28, 28, 28, 28, 28, 36, 36, 36, 36, 36, 36, 36, 36, 45, 45, 45, 45, 45, 45, 45, 45, 45, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66

Offset: 1

Author

Gary W. Adamson, Jan 27 2007

Keywords

• nonn
• tabl

Comments

Seen as a sequence, these are the triangular numbers applied to the Kruskal-Macaulay function A123578. - Peter Luschny, Oct 29 2022

Examples

			First few rows of the triangle are:
   1;
   3,  3;
   6,  6,  6;
  10, 10, 10, 10;
  15, 15, 15, 15, 15;
  ...
		

Links

• Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
• Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A000217, A000384, A002024, A002411 (row sums), A003057, A057944, A123578.

Programs

Haskell

a127739 n k = a127739_tabl !! (n-1) !! (k-1)
a127739_row n = a127739_tabl !! (n-1)
a127739_tabl = zipWith ($) (map replicate [1..]) $ tail a000217_list
-- Reinhard Zumkeller, Feb 03 2012, Mar 18 2011

Maple

A127739 := proc(n) local t, s; t := 1; s := 0;
while t <= n do s := s + 1; t := t + s od; s*(1 + s)/2 end:
seq(A127739(n), n = 1..66); # Peter Luschny, Oct 29 2022

Mathematica

Table[n(n+1)/2,{n,100},{n}]//Flatten (* Zak Seidov, Mar 19 2011 *)

PARI

A127739=n->binomial((sqrtint(8*n)+3)\2,2) \\ M. F. Hasler, Mar 09 2014

Python

from math import isqrt
def A127739(n): return (r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(r+1)>>1 # Chai Wah Wu, Nov 07 2024

Formula

Central terms: T(2*n-1,n) = A000384(n). - Reinhard Zumkeller, Mar 18 2011

a(n) = A003057(n)*A002024(n)/2; a(n) = (t+2)*(t+1)/2, where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 08 2013

Sum_{n>=1} 1/a(n)^2 = 8 - 2*Pi^2/3. - Amiram Eldar, Aug 15 2022

a(n) = k(n)*(1 + k(n))/2 = A000217(A123578(n)), where k = A123578. - Peter Luschny, Oct 29 2022

Extensions

Name edited by Michel Marcus, Apr 30 2020