A368842 -id:A368842 - cp's OEIS frontend

A368843 a(n) gives the number of triples of equally spaced 1's in the binary expansion of n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 2, 2, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 2, 2, 3, 4, 6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 3, 0, 0, 0, 0, 1, 2, 1

Offset: 0

Rémy Sigrist, Jan 07 2024

			For n = 277:
- the binary expansion of 277 is "100010101",
- we have the following triples:  1   1   1
                                      1 1 1
- so a(277) = 2.

Index entries for sequences related to binary expansion of n

Cf. A368842, A368844.

a(n, t = 1, base = 2) = { my (d = digits(n, base), v = 0); for (i = 1, #d-2, if (d[i]==t, forstep (j = i+2, #d, 2, if (d[i]==d[j] && d[i]==d[(i+j)/2], v++;);););); return (v); }

def A368843(n):
    l = len(s:=bin(n)[2:])
    return sum(1 for i in range(l-2) for j in range(1,l-i+1>>1) if s[i:i+(j<<1)+1:j]=='111') # Chai Wah Wu, Jan 10 2024

a(2*n) = a(n).

a(2*n + 1) >= a(n).

A368844 a(n) gives the number of triples of equally spaced 0's in the binary expansion of n (without leading zeros).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 2, 2, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6, 4, 3, 2, 2, 2, 1, 1, 2, 0, 1, 0, 0, 0, 0, 0, 3, 2, 0, 0, 1, 1, 0

Offset: 0

Rémy Sigrist, Jan 07 2024

			For n = 277:
- the binary expansion of 277 is "100010101",
- we have the following triples:   000
                                   0 0 0
                                     0 0 0
- so a(277) = 3.

Index entries for sequences related to binary expansion of n

Cf. A368842, A368843.

a(n, t = 0, base = 2) = { my (d = digits(n, base), v = 0); for (i = 1, #d-2, if (d[i]==t, forstep (j = i+2, #d, 2, if (d[i]==d[j] && d[i]==d[(i+j)/2], v++;);););); return (v); }

def A368844(n):
    l = len(s:=bin(n)[3:])
    return sum(1 for i in range(l-2) for j in range(1,l-i+1>>1) if s[i:i+(j<<1)+1:j]=='000') # Chai Wah Wu, Jan 10 2024

a(2*n) >= a(n).

a(2*n + 1) = a(n).

A368841 Nonnegative integers whose binary expansions (without leading zeros) have no three equally spaced equal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 22, 25, 26, 27, 36, 37, 38, 41, 44, 45, 50, 51, 52, 54, 76, 82, 83, 90, 100, 101, 102, 108, 153, 165, 204

Offset: 1

Rémy Sigrist, Jan 07 2024

Also numbers k such that A368842(k) = 0.
Also numbers k such that A368857(k) < 3.
This sequence is finite by Van der Waerden's theorem.

Cf. A005346, A368842, A368857.

is(n, base = 2) = { my (d = digits(n, base)); for (i = 1, #d-2, forstep (j = i+2, #d, 2, if (d[i]==d[j] && d[i]==d[(i+j)/2], return (0);););); return (1); }

cp's OEIS Frontend

A368843 a(n) gives the number of triples of equally spaced 1's in the binary expansion of n.

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A368844 a(n) gives the number of triples of equally spaced 0's in the binary expansion of n (without leading zeros).

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PARI

Python

Formula

A368841 Nonnegative integers whose binary expansions (without leading zeros) have no three equally spaced equal digits.

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PARI