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A181388 a(n) = Sum_{k=1..n} 2^T(k-1), where T = A000217 are the triangular numbers 0, 1, 3, 6, 10, ... . For n=0 we have the empty sum equal to 0.

%I A181388 #20 Jan 17 2020 10:45:28
%S A181388 0,1,3,11,75,1099,33867,2131019,270566475,68990043211,35253362132043,
%T A181388 36064050381096011,73823040345219302475,302305277944002512979019,
%U A181388 2476182383848704552311227467,40567295389687189552446813799499,1329268563080305560093359507094144075
%N A181388 a(n) = Sum_{k=1..n} 2^T(k-1), where T = A000217 are the triangular numbers 0, 1, 3, 6, 10, ... . For n=0 we have the empty sum equal to 0.
%C A181388 Original definition: The binary representation of each integer in the sequence consists of a single leading bit, followed by a string of n-1 zeros, followed by the previous integer. i.e. 3 = 2^1 + 1, 11 = 2^(2+1) + 3, 75 = 2^(3+2+1) + 11, and so on.
%C A181388 Numbers in this sequence may be used as a multiplier in hash functions to scatter and interleave bits.
%H A181388 Harvey P. Dale, <a href="/A181388/b181388.txt">Table of n, a(n) for n = 0..82</a>
%F A181388 a(n) = Sum_{k=1..n} A006125(k). - _R. J. Mathar_, Oct 18 2010
%F A181388 a(n) = a(n-1) + 2*(a(n-1) - a(n-2))^2/(a(n-2) - a(n-3)) for n >= 3. - _Robert Israel_, Aug 28 2014
%p A181388 f := proc(n) option remember; f(n-1) + 2^(ilog2(f(n-1))+ n - 1); end proc:
%p A181388 f(0) := 0:f(1):= 1:
%p A181388 seq(f(n),n=0..60); # updated by _Robert Israel_, Aug 28 2014
%t A181388 Join[{0},Accumulate[2^Accumulate[Range[0,15]]]] (* _Harvey P. Dale_, Mar 10 2016 *)
%o A181388 (PARI) a(n)=sum(k=1,n,2^(k*(k-1)/2)) \\ _M. F. Hasler_, Aug 28 2014
%Y A181388 Cf. A000217, A006125.
%K A181388 base,nonn
%O A181388 0,3
%A A181388 _Roman Pearce_, Oct 17 2010
%E A181388 Prefixed initial term a(0)=0 and simplified definition - _M. F. Hasler_, Aug 28 2014