A144328 A002260 preceded by a column of 1's: a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Offset: 1

Author

Gary W. Adamson, Sep 18 2008

Keywords

• nonn
• tabl

Comments

Row sums = A000124.

Eigensequence of the triangle = A000142, the factorials.

The triangle as an infinite lower triangular matrix * [1,2,3,...] = A064999.

Generated from A128227 by rotating each row by one position to the right. - R. J. Mathar, Sep 25 2008

A sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A144328 is the reluctant sequence of A028310 (1 followed by the natural numbers). - Boris Putievskiy, Dec 12 2012

If offset were changed to 0, a(n) would equal the

Let S_n be the set of partitions of n into distinct parts where the number of parts is maximal for that n. For example, for n=6, the set S_6 consists of just one such partition: S_6={1,2,3}. Similarly, for n=7, S_7={1,2,4}, But for n=8, S_8 will contain two partitions S_8= { {1,2,5}, {1,3,4} }. Then |S(n)| = a(n+1). Cf. A178702. - David S. Newman and Benoit Jubin, Dec 13 2010

Examples

			First few rows of the triangle:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3;
  1, 1, 2, 3, 4;
  1, 1, 2, 3, 4, 5;
  ...
		

Links

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

Crossrefs

Cf. A000124, A000142, A002260, A010054, A064999, A028310, A156702.

Programs

Haskell

a144328 n k = a144328_tabl !! (n-1) !! (k-1)
a144328_row n = a144328_tabl !! (n-1)
a144328_tabl = [1] : map (\xs@(x:_) -> x : xs) a002260_tabl
-- Reinhard Zumkeller, Apr 29 2015

Mathematica

Flatten[Table[Join[{1},Range[n]],{n,0,11}]] (* Harvey P. Dale, Aug 10 2013 *)

Python

from math import comb, isqrt
def A144328(n): return n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2)-(comb(isqrt(n-1<<1)+1,2)!=n-1) # Chai Wah Wu, Nov 08 2024

Formula

Triangle A002260 (natural numbers crescendo triangle) preceded by a column of 1's, = a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.

a(n) = A028310(m-1), where m = n-t*(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012

a(n) = A002260(n)+A010054(n-1)-1. - Chai Wah Wu, Nov 08 2024