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A280873 Numbers whose binary expansion does not begin 10 and do not contain 2 adjacent 0's; Ahnentafel numbers of X-chromosome inheritance of a male. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The number of ancestors at generation n from whom a living individual may have received an X chromosome allele is Fn, the n-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

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

PROG

(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).

CROSSREFS

Cf. A003714, A054429.

Intersection of A003754 and A004760.

Positions where A163511 obtains squarefree (A005117) values.

Cf. also A293437 (a subsequence).

Sequence in context: A333002 A175048 A294231 * A293437 A282354 A088146

Adjacent sequences:  A280870 A280871 A280872 * A280874 A280875 A280876

KEYWORD

nonn,base

AUTHOR

Floris Strijbos, Jan 09 2017

EXTENSIONS

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

STATUS

approved

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Last modified April 7 04:20 EDT 2020. Contains 333292 sequences. (Running on oeis4.)