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%I #24 Feb 16 2024 14:31:03
%S 0,1,5,37,549,16933,1065509,135283237,34495021605,17626681066021,
%T 18032025190548005,36911520172609651237,151152638972001256489509,
%U 1238091191924352276155613733,20283647694843594776223406899749,664634281540152780046679753547072037
%N a(n) = Sum_{k=1..n} 2^(T(k)-1), where T(k)=k(k+1)/2 = A000217(k).
%C Similar to A181388, this occurs as binary encoding of a straight n-omino lying on the y-axis, when the grid points of the first quadrant (N x N, N={0,1,2,...}) are given the weight 2^k, with k=0, 1,2, 3,4,5, ... filled in by antidiagonals.
%C Numbers k such that the k-th composition in standard order (row k of A066099) is a reversed initial interval. - _Gus Wiseman_, Apr 02 2020
%e Label the cells of an infinite square matrix with 0,1,2,3,... along antidiagonals:
%e 0 1 3 6 10 ...
%e 2 4 7 ...
%e 5 8 ...
%e 9 ...
%e ....
%e Now any subset of these cells can be represented by the sum of 2 raised to the power written in the given cells. In particular, the subset consisting of the first cell in the first 1, 2, 3, ... rows is represented by 2^0, 2^0+2^2, 2^0+2^2+2^5, ...
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t Select[Range[0,1000],normQ[stc[#]]&&Greater@@stc[#]&] (* _Gus Wiseman_, Apr 02 2020 *)
%o (PARI) t=0;vector(20,n,t+=2^(n*(n+1)/2-1)) \\ yields the vector starting with a[1]=1
%o (PARI) t=0;vector(20,n,if(n>1,t+=2^(n*(n-1)/2-1))) \\ yields the vector starting with 0
%o (Python)
%o a = 0
%o for n in range(1,17): print(a, end =', '); a += 1<<(n-1)*(n+2)//2 # _Ya-Ping Lu_, Jan 23 2024
%Y The version for prime (rather than binary) indices is A002110.
%Y The non-strict generalization is A114994.
%Y The non-reversed version is A164894.
%Y Intersection of A333256 and A333217.
%Y Partial sums of A036442.
%Y Cf. A000120, A029931, A048793, A066099, A070939, A124766, A228351, A233564, A272919, A333218.
%K nonn
%O 0,3
%A _M. F. Hasler_, Aug 28 2014
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