A004780 -id:A004780 - cp's OEIS frontend

A365538 a(0) = 1; otherwise, for i >= 0, a(4i+0) = a(4i+1) = a(2i), a(4i+2) = 2*a(2i+1), a(4i+3) = 0.

1, 1, 2, 0, 2, 2, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0, 2, 2, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 0, 4, 4, 0, 0, 4, 4, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Roland Kneer, Oct 23 2023

Related to a model of X-chromosome inheritance:

The two X chromosomes of a female are inherited one from each parent, while the X chromosome of a male is always inherited from his mother. Thus, the probability distribution of inheritance from the parents (mother, father) is (0.5, 0.5) for a female and (1, 0) for a male. For the inheritance of any X-chromosome of a female from the 2^i ancestors of the i-th generation before (right to left on an ahnentafel), the distribution is given by the first 2^i terms of the sequence, divided by 2^i. For example, the X-chromosome of a man, which was inherited from his mother, was inherited from his mother's 16 great-great-grandparents with probabilities 1/16, 1/16, 1/8, 0, 1/8, 1/8, 0, 0, 1/8, 1/8, 1/4, 0, 0, 0, 0, 0.

Roland Kneer, Table of n, a(n) for n = 0..10000

Cf. A280873, A003754.

Positions of 0's: A004780 (complement of A003714).

This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:
  1;
  1;
  2, 0;
  2, 2, 0, 0;
  2, 2, 4, 0, 0, 0, 0, 0;
  2, 2, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...

A300302 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation contains the binary representation of n as a substring.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 5, 4, 5, 8, 7, 6, 5, 6, 10, 9, 11, 8, 6, 7, 12, 11, 12, 12, 9, 7, 8, 14, 13, 13, 16, 13, 10, 8, 9, 16, 15, 14, 20, 17, 14, 11, 9, 10, 18, 17, 23, 22, 21, 18, 15, 12, 10, 11, 20, 19, 24, 28, 24, 22, 19, 19, 13, 11, 12, 22, 21, 25, 32, 29

Offset: 1

Rémy Sigrist, Mar 08 2018

Each positive number k appears A122953(k) times in this array.

			Square array begins:
  n\k|    1    2    3    4    5    6    7    8    9   10
  ---+--------------------------------------------------
    1|    1    2    3    4    5    6    7    8    9   10  <--  A000027
    2|    2    4    5    6    8    9   10   11   12   13  <--  A062289
    3|    3    6    7   11   12   13   14   15   19   22  <--  A004780
    4|    4    8    9   12   16   17   18   19   20   24  <--  A004753
    5|    5   10   11   13   20   21   22   23   26   27  <--  A004748
    6|    6   12   13   14   22   24   25   26   27   28  <--  A004749
    7|    7   14   15   23   28   29   30   31   39   46  <--  A004781
    8|    8   16   17   24   32   33   34   35   40   48  <--  A004779
    9|    9   18   19   25   36   37   38   39   41   50
   10|   10   20   21   26   40   41   42   43   52   53  <--  A132782

Rémy Sigrist, Perl program for A300302

Cf. A000027, A004748, A004749, A004753, A004779, A004780, A004781, A062289, A122953, A132782.

Perl
```
See Links section.
```

T(n, 1) = n.
T(n, 2) = 2*n.
T(n, 3) = 2*n + 1.
T(1, n) = A000027(n).
T(2, n) = A062289(n).
T(3, n) = A004780(n).
T(4, n) = A004753(n).
T(5, n) = A004748(n).
T(6, n) = A004749(n).
T(7, n) = A004781(n).
T(8, n) = A004779(n-1).
T(10, n) = A132782(n).

A348556 Binary expansion contains 4 adjacent 1's.

Original entry on oeis.org

15, 30, 31, 47, 60, 61, 62, 63, 79, 94, 95, 111, 120, 121, 122, 123, 124, 125, 126, 127, 143, 158, 159, 175, 188, 189, 190, 191, 207, 222, 223, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 271, 286, 287, 303, 316, 317, 318

Offset: 1

Charles R Greathouse IV, Oct 22 2021

For k > 0, each term m = 2^(k+3) - 1 is the end of a run of A083593(k-1) consecutive terms. For k = 4, from a(13) = 120 up to a(20) = 2^7-1 = 127, there are A083593(3) = 8 consecutive terms corresponding to 1111000, 1111001, 1111010, 1111011, 1111100, 1111101, 111110 and 1111111. - Bernard Schott, Feb 20 2022

Michel Marcus, Table of n, a(n) for n = 1..10000
Index entries for 2-automatic sequences.

Binary expansion contains k adjacent 1s: A000027 (1), A004780 (2), A004781 (3), this sequence (4).
Cf. A006520, A083593.
Subsequences: A110286, A195744.

q:= n-> verify([1$4], Bits[Split](n), 'sublist'):
select(q, [$0..400])[];  # Alois P. Heinz, Oct 22 2021

Select[Range[300], StringContainsQ[IntegerString[#, 2], "1111"] &] (* Amiram Eldar, Oct 22 2021 *)

is(n)=n=bitand(n,n<<2); !!bitand(n,n<<1);

def ok(n): return "1111" in bin(n)
print([k for k in range(319) if ok(k)]) # Michael S. Branicky, Oct 22 2021

a(n) ~ n.

a(n+1) <= a(n) + 16.

cp's OEIS Frontend

A365538 a(0) = 1; otherwise, for i >= 0, a(4i+0) = a(4i+1) = a(2i), a(4i+2) = 2*a(2i+1), a(4i+3) = 0.

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A300302 Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation contains the binary representation of n as a substring.

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A348556 Binary expansion contains 4 adjacent 1's.

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