A058052 Sum of the distances between the 2^n vertices in the De Bruijn Graphs on words of length n on alphabet {0,1}.

2, 18, 118, 680, 3620, 18274, 88760, 418900, 1933904, 8775534, 39277136, 173843142, 762388102, 3317784992, 14344443516, 61671799608, 263865053452, 1124175400716, 4771570406736, 20185774001256, 85141101913670

Offset

1,1

Comments

Given two words X,Y in {0,1}^N, the distance d(X,Y) is the least integer K such that there exists a word M with X=UM and Y=MV and |U|=|V|=K. Define a(N)=sum(d(X,Y); X,Y in {0,1}^N).

Links

Table of n, a(n) for n=1..21.

Eric Weisstein's World of Mathematics, de Bruijn Sequence.

Example

d(0,0)=0, d(0,1)=1, d(1,0)=1, d(1,1)=0, hence a(1)=0+1+1+0=2.
d(00,00)=0, d(00,01)=1, d(00,10)=2, d(00,11)=2, d(01,00)=2, d(01,01)=0, d(01,10)=1, d(01,11)=1, d(10,00)=1, d(10,01)=1, d(10,10)=0, d(10,11)=2, d(11,00)=2, d(11,01)=2, d(11,10)=1, d(11,11)=0, hence a(2)=0+1+2+2+2+0+1+1+1+1+0+2+2+2+1+0=18.

Prog

(Python)
from numba import njit
@njit
def d(x, y, n):
  for k in range(n):
    mask = (1 << (n-k)) - 1
    if x & mask == (y >> k): return k
  return n
@njit
def a(n):
  s = 0
  for x in range(2**(n-1)):
    for y in range(2**n):
      s += d(x, y, n)
  return 2*s
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Feb 18 2021

Crossrefs

Cf. A166316.

Sequence in context: A027433 A153338 A007798 * A119578 A052610 A052653

Adjacent sequences: A058049 A058050 A058051 * A058053 A058054 A058055

Keyword

nonn

Author

Serge Burckel (burckel(AT)iml.univ-mrs.fr), Nov 19 2000

Extensions

a(9)-a(21) from Michael S. Branicky, Feb 18 2021

Status

approved