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

1,2

Comments

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

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.

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A262871 A160745 A105676 * A327142 A175394 A070318

Adjacent sequences: A127736 A127737 A127738 * A127740 A127741 A127742

Keyword

nonn,tabl

Author

Gary W. Adamson, Jan 27 2007

Extensions

Name edited by Michel Marcus, Apr 30 2020

Status

approved