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.

0, 1, 3, 11, 75, 1099, 33867, 2131019, 270566475, 68990043211, 35253362132043, 36064050381096011, 73823040345219302475, 302305277944002512979019, 2476182383848704552311227467, 40567295389687189552446813799499, 1329268563080305560093359507094144075

Offset: 0

Author

Roman Pearce, Oct 17 2010

Keywords

• base
• nonn

Comments

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.

Numbers in this sequence may be used as a multiplier in hash functions to scatter and interleave bits.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..82

Crossrefs

Cf. A000217, A006125.

Programs

Maple

f := proc(n) option remember; f(n-1) + 2^(ilog2(f(n-1))+ n - 1); end proc:
f(0) := 0:f(1):= 1:
seq(f(n),n=0..60); # updated by Robert Israel, Aug 28 2014

Mathematica

Join[{0},Accumulate[2^Accumulate[Range[0,15]]]] (* Harvey P. Dale, Mar 10 2016 *)

PARI

a(n)=sum(k=1,n,2^(k*(k-1)/2)) \\ M. F. Hasler, Aug 28 2014

Formula

a(n) = Sum_{k=1..n} A006125(k). - R. J. Mathar, Oct 18 2010

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

Extensions

Prefixed initial term a(0)=0 and simplified definition - M. F. Hasler, Aug 28 2014