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

%I A137688 #37 Feb 10 2023 14:39:18
%S A137688 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,
%T A137688 64,64,128,128,128,128,128,128,128,128,256,256,256,256,256,256,256,
%U A137688 256,256,512,512,512,512,512,512,512,512,512,512,1024,1024,1024,1024,1024,1024
%N A137688 2^A003056: 2^n appears n+1 times.
%C A137688 First differences of A007664.
%C A137688 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
%C A137688 From _Paul Curtz_, Oct 23 2018: (Start)
%C A137688 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
%C A137688    1     1/2   1/4   1/8 ...
%C A137688   -1/2  -1/4  -1/8  -1/16 ...
%C A137688    1/4   1/8   1/16  1/32 ...
%C A137688   -1/8  -1/16 -1/32 -1/64 ...
%C A137688 etc.
%C A137688 The denominators on the antidiagonals are a(n). (End)
%H A137688 Vincenzo Librandi, <a href="/A137688/b137688.txt">Rows n = 0..100, flattened</a>
%F A137688 a(n) = 2^[sqrt(2n+2)-.5] = 2^A003056(n) = A007664(n+1) - A007664(n).
%F A137688 Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 19 2013
%F A137688 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
%F A137688 Sum_{n>=0} 1/a(n) = 4. - _Amiram Eldar_, Aug 16 2022
%e A137688 Triangle T(n,k) begins:
%e A137688 1
%e A137688 2, 2
%e A137688 4, 4, 4
%e A137688 8, 8, 8, 8
%e A137688 16, 16, 16, 16, 16
%e A137688 32, 32, 32, 32, 32, 32
%e A137688 64, 64, 64, 64, 64, 64, 64
%e A137688 - _Philippe Deléham_, Dec 26 2013
%p A137688 seq(seq(2^n,k=1..n+1),n=0..10); # _Muniru A Asiru_, Oct 23 2018
%t A137688 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 *)
%o A137688 (PARI) A137688(n)= 1<<floor(sqrt(2*n+2)-.5)
%o A137688 (Haskell)
%o A137688 a137688 n = a137688_list !! n
%o A137688 a137688_list = concat $ zipWith ($) (map replicate [1..]) (map (2^) [0..])
%o A137688 -- _Reinhard Zumkeller_, Mar 18 2011
%o A137688 (GAP) Flat(List([0..10],n->List([1..n+1],k->2^n))); # _Muniru A Asiru_, Oct 23 2018
%o A137688 (Python)
%o A137688 from math import isqrt
%o A137688 def A137688(n): return 1<<(isqrt((n<<3)+1)-1>>1) # _Chai Wah Wu_, Feb 10 2023
%Y A137688 Cf. A003056, A007664 (gives partial sums).
%K A137688 easy,nonn,tabl
%O A137688 0,2
%A A137688 _M. F. Hasler_, Feb 06 2008