A140531 -id:A140531 - cp's OEIS frontend

A059268 Concatenate subsequences [2^0, 2^1, ..., 2^n] for n = 0, 1, 2, ...

1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 32, 1, 2, 4, 8, 16, 32, 64, 1, 2, 4, 8, 16, 32, 64, 128, 1, 2, 4, 8, 16, 32, 64, 128, 256, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048

Offset: 0

N. J. A. Sloane, Jan 23 2001

Triangular array T(n,k) read by rows, where T(n,k) = i!*j! times coefficient of x^n*y^k in exp(x+2y).
T(n,k) is the number of subsets of {0,1,...,n} whose largest element is k. To see this, let A be any subset of the 2^k subsets of {0,1,...,k-1}. Then there are 2^k subsets of the form (A U {k}). See example below. - Dennis P. Walsh, Nov 27 2011
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. A059268 is reluctant sequence of sequence A000079. - Boris Putievskiy, Dec 17 2012

			T(4,3)=8 since there are 8 subsets of {0,1,2,3,4} whose largest element is 3, namely, {3}, {0,3}, {1,3}, {2,3}, {0,1,3}, {0,2,3}, {1,2,3}, and {0,1,2,3}.
Triangle starts:
  1;
  1, 2;
  1, 2, 4;
  1, 2, 4, 8;
  1, 2, 4, 8, 16;
  1, 2, 4, 8, 16, 32;
  ...

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A140531.
Cf. A000079.
Cf. A131816.
Row sums give A126646.

a059268 n k = a059268_tabl !! n !! k
a059268_row n = a059268_tabl !! n
a059268_tabl = iterate (scanl (+) 1) [1]
-- Reinhard Zumkeller, Apr 18 2013, Jul 05 2012

Maple
```
seq(seq(2^k,k=0..n),n=0..10);
```

Table[2^k, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 10 2013 *)

from math import isqrt
def A059268(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return 1<>1) # Chai Wah Wu, Feb 24 2025

E.g.f.: exp(x+2*y) (T coordinates).
a(n) = A018900(n+1) - A140513(n). - Reinhard Zumkeller, Jun 24 2009
T(n,k) = A173786(n-1,k-1) - A173787(n-1,k-1), 0Reinhard Zumkeller, Feb 28 2010
T(n,k) = 2^k. - Reinhard Zumkeller, Jan 29 2010
As a linear array, the sequence is a(n) = 2^(n-1-t*(t+1)/2), where t = floor((-1+sqrt(8*n-7))/2), n>=1. - Boris Putievskiy, Dec 17 2012
As a linear array, the sequence is a(n) = 2^(n-1-t*(t+1)/2), where t = floor(sqrt(2*n)-1/2), n>=1. - Zhining Yang, Jun 09 2017

Formula corrected by Reinhard Zumkeller, Feb 23 2010

A140648 Triangle T(n,m) which can create A140642 without help of Jacobsthal numbers.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 8, 2, 0, 1, 16, 4, 1, 0, 2, 32, 8, 2, 0, 1, 4, 64, 16, 4, 1, 0, 2, 8, 128, 32, 8, 2, 0, 1, 4, 16, 256, 64, 16, 4, 1, 0, 2, 8, 32, 512, 128, 32, 8, 2, 0, 1, 4, 16, 64

Offset: 0

Paul Curtz, Jul 09 2008

This triangle T(.,.) provides the additional terms if A140642 is constructed with a Pascal-type recurrence: A140642(n+1,m+1) = A140642(n,m) + A140642(n,m+1) + T(n,m+1).
Note almost odd palindromes (of squares) followed by their double.
Examples: 40=16+20+4, 42=20+21+1, 43=21+22+0, 44=22+24+2.

			Triangle begins:
    1;
    2,  0;
    4,  1,  0;
    8,  2,  0,  1;
   16,  4,  1,  0,  2;
   32,  8,  2,  0,  1,  4;
   64, 16,  4,  1,  0,  2,  8;
  128, 32,  8,  2,  0,  1,  4, 16;

Cf. A083329 (row sums).

Southeast diagonals based on A131577 (which is also in A140531). First preceded with 1, 0. Second with 2, 1, 0. Tends towards even palindromes, second part being A131577. Verticals: A000079, A131577, (0, A131577), ... .

cp's OEIS Frontend

A059268 Concatenate subsequences [2^0, 2^1, ..., 2^n] for n = 0, 1, 2, ...

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