A280873 Numbers whose binary expansion does not begin 10 and does not contain 2 adjacent 0's; Ahnentafel numbers of X-chromosome inheritance of a male.

0, 1, 3, 6, 7, 13, 14, 15, 26, 27, 29, 30, 31, 53, 54, 55, 58, 59, 61, 62, 63, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 213, 214, 215, 218, 219, 221, 222, 223, 234, 235, 237, 238, 239, 245, 246, 247, 250, 251, 253, 254, 255

Offset: 0

Author

Floris Strijbos, Jan 09 2017

Keywords

• nonn
• base
• easy

Comments

The number of ancestors at generation m from whom a living individual may have received an X chromosome allele is F_m, the m-th term of the Fibonacci Sequence.

From Antti Karttunen, Oct 11 2017: (Start)

The starting offset is zero (with a(0) = 0) for the same reason that we have A003714(0) = 0. Indeed, b(n) = A054429(A003714(n)) for n >= 0 yields the terms of this sequence, but in different order.

A163511(a(n)) for n >= 0 gives a permutation of squarefree numbers (A005117). See also A277006.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10946
• David Eppstein, Self-recursive generators (Python recipe)
• L. A. D. Hutchison, N. M. Myres and S. R. Woodward, Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships, Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04, Colorado Springs), pp. 42-49, Sept. 2004.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A003714, A054429, A365538.

Intersection of A003754 and A004760.

Positions where A163511 obtains squarefree (A005117) values.

Cf. also A293437 (a subsequence).

Programs

Maple

gen[0]:= {0,1,3}:
gen[1]:= {6,7}:
for n from 2 to 10 do
  gen[n]:= map(t -> 2*t+1, gen[n-1]) union
      map(t -> 2*t, select(type, gen[n-1],odd))
od:
sort(convert(`union`(seq(gen[i],i=0..10)),list)); # Robert Israel, Oct 11 2017

Mathematica

male = {1, 3}; generations = 8;
Do[x = male[[i - 1]]; If[EvenQ[x],
                          male = Append[ male,   2*x + 1] ,
                          male = Flatten[Append[male, {2*x, 2*x + 1}]]]
       , {i, 3, Fibonacci[generations + 1]}]; male

PARI

isA003754(n) = { n=bitor(n, n>>1)+1; n>>=valuation(n, 2); (n==1); }; \\ After Charles R Greathouse IV's Feb 06 2017 code.
isA004760(n) = (n<2 || (binary(n)[2])); \\ This function also from Charles R Greathouse IV, Sep 23 2012
isA280873(n) = (isA003754(n) && isA004760(n));
n=0; k=0; while(k <= 10946, if(isA280873(n),write("b280873.txt", k, " ", n);k=k+1); n=n+1;); \\ Antti Karttunen, Oct 11 2017

Python

def A280873():
    yield 1
    for x in A280873():
        if ((x & 1) and (x > 1)):
            yield 2*x
        yield 2*x+1
def take(n, g):
  '''Returns a list composed of the next n elements returned by generator g.'''
  z = []
  if 0 == n: return(z)
  for x in g:
    z.append(x)
    if n > 1: n = n-1
    else: return(z)
take(120, A280873())
# Antti Karttunen, Oct 11 2017, after the given Mathematica-code (by Floris Strijbos) and a similar generator-example for A003714 by David Eppstein (cf. "Self-recursive generators" link).

Formula

{a(n) : n >= 1} = {k >= 1 : A365538(A054429(k)) > 0}. - Peter Munn, Jan 22 2024

Extensions

a(0) = 0 prepended and more descriptive alternative name added by Antti Karttunen, Oct 11 2017