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A137688 2^A003056: 2^n appears n+1 times. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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)

LINKS

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

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

EXAMPLE

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

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 *)

PROG

(PARI) A137688(n)= 1<<floor(sqrt(2*n+2)-.5)

(Haskell)

a137688 n = a137688_list !! n

a137688_list = concat $ zipWith ($) (map replicate [1..]) (map (2^) [0..])

-- Reinhard Zumkeller, Mar 18 2011

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

CROSSREFS

Cf. A003056, A007664 (gives partial sums).

Sequence in context: A320197 A124340 A071165 * A033720 A033728 A033744

Adjacent sequences:  A137685 A137686 A137687 * A137689 A137690 A137691

KEYWORD

easy,nonn,tabl

AUTHOR

M. F. Hasler, Feb 06 2008

STATUS

approved

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Last modified October 17 18:58 EDT 2019. Contains 328127 sequences. (Running on oeis4.)