A137688 2^A003056: 2^n appears n+1 times.

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

Offset: 0

Author

M. F. Hasler, Feb 06 2008

Keywords

• easy
• nonn
• tabl

Comments

First differences of A007664.

Viewed as a triangle, it is computed like Pascal's triangle, but with 2^n on the triangle edges. - T. D. Noe, Jul 31 2013

From Paul Curtz, Oct 23 2018: (Start)

Oresme numbers O(n) = n/2^n are an autosequence of the first kind. The corresponding sequence of the second kind is 1/2^n. The difference table is

1 1/2 1/4 1/8 ...

-1/2 -1/4 -1/8 -1/16 ...

1/4 1/8 1/16 1/32 ...

-1/8 -1/16 -1/32 -1/64 ...

etc.

The denominators on the antidiagonals are a(n). (End)

Examples

			Triangle T(n,k) begins:
1
2, 2
4, 4, 4
8, 8, 8, 8
16, 16, 16, 16, 16
32, 32, 32, 32, 32, 32
64, 64, 64, 64, 64, 64, 64
- _Philippe Deléham_, Dec 26 2013
		

Links

• Vincenzo Librandi, Rows n = 0..100, flattened

Crossrefs

Cf. A003056, A007664 (gives partial sums).

Programs

GAP

Flat(List([0..10],n->List([1..n+1],k->2^n))); # Muniru A Asiru, Oct 23 2018

Haskell

a137688 n = a137688_list !! n
a137688_list = concat $ zipWith ($) (map replicate [1..]) (map (2^) [0..])
-- Reinhard Zumkeller, Mar 18 2011

Maple

seq(seq(2^n,k=1..n+1),n=0..10); # Muniru A Asiru, Oct 23 2018

Mathematica

t = {}; Do[r = {}; Do[If[k == 0||k == n, m = 2^n, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 9}]; t = Flatten[t] (* Vincenzo Librandi, Aug 01 2013 *)

PARI

A137688(n)= 1<
				

Python

from math import isqrt
def A137688(n): return 1<<(isqrt((n<<3)+1)-1>>1) # Chai Wah Wu, Feb 10 2023

Formula

a(n) = 2^[sqrt(2n+2)-.5] = 2^A003056(n) = A007664(n+1) - A007664(n).

Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

Viewed as a triangle T(n,k) : T(n,k)=2*T(n-1,k)+2*T(n-1,k-1)-4*T(n-2,k-1), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 26 2013

Sum_{n>=0} 1/a(n) = 4. - Amiram Eldar, Aug 16 2022