A121016: Numbers Whose Binary Expansion Is Properly Periodic. Or A328594: Numbers Whose Binary Expansion Is Aperiodic

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A121016: Numbers whose binary expansion is properly periodic. or A328594: Numbers whose binary expansion is aperiodic

In the realm of number theory and binary expansion, there exist two distinct categories of numbers: those with properly periodic binary expansions and those with aperiodic binary expansions. The former category is represented by the sequence A121016, while the latter is represented by the sequence A328594. In this article, we will delve into the world of binary expansions, exploring the properties and characteristics of these two types of numbers.

Properly Periodic Binary Expansions

A properly periodic binary expansion is a sequence of binary digits (0s and 1s) that repeats indefinitely in a predictable pattern. This means that the sequence will eventually repeat itself, with the same sequence of digits appearing over and over again. For example, the binary expansion of the number 1/3 is 0.010101... , which is a properly periodic sequence.

Properties of Properly Periodic Binary Expansions

Properly periodic binary expansions have several interesting properties. One of the most notable is that they can be represented as a finite sum of powers of 2. This is because the repeating pattern of the sequence can be expressed as a sum of terms, each of which is a power of 2 multiplied by a coefficient.

For example, the binary expansion of 1/3 can be represented as:

1/3 = 1/2^1 + 1/2^3 + 1/2^5 + ...

This representation shows that the binary expansion of 1/3 is a sum of powers of 2, with each term being a power of 2 multiplied by a coefficient.

Aperiodic Binary Expansions

An aperiodic binary expansion, on the other hand, is a sequence of binary digits that does not repeat indefinitely in a predictable pattern. This means that the sequence will not eventually repeat itself, and the same sequence of digits will not appear over and over again.

Properties of Aperiodic Binary Expansions

Aperiodic binary expansions have several interesting properties as well. One of the most notable is that they can be represented as a non-terminating sum of powers of 2. This is because the sequence does not repeat indefinitely, and the terms of the sum will continue to appear indefinitely.

For example, the binary expansion of the number e (the base of the natural logarithm) is an aperiodic sequence:

e = 2.71828... (in binary)

This representation shows that the binary expansion of e is a non-terminating sum of powers of 2.

Code Golf and Sequence

In the context of code golf and sequence, the study of properly periodic and aperiodic binary expansions can be a fascinating topic. By analyzing the properties of these sequences, we can gain insights into the underlying structure of binary expansions and develop new techniques for compressing and representing these sequences.

Compressing Dis Programs

One of the possible applications of this knowledge is in compressing Dis programs, which are a type of program that uses only the characters } and { (and optionally _) to represent the program. By analyzing the properties of properly periodic and aperiodic binary expansions, we can develop new techniques for compressing these programs and representing them in a more compact form.

Example: Compressing Dis Programs with Only } and { Characters

For example, consider the following Dis program:

} { } { } { } { } { } { } { }

This program can be compressed using the properties of properly periodic binary expansions. By analyzing the sequence of } and { characters, we can identify a repeating pattern and represent the program as a sum of powers of 2.

For example, the program can be represented as:

} { } { } { } { } { } { } { } { } = 1/2^1 + 1/2^3 + 1/2^5 + ...

This representation shows that the program can be compressed into a more compact form using the properties of properly periodic binary expansions.

Conclusion

In conclusion, the study of properly periodic and aperiodic binary expansions is a fascinating topic that has many applications in code golf and sequence. By analyzing the properties of these sequences, we can gain insights into the underlying structure of binary expansions and develop new techniques for compressing and representing these sequences.

References

  • A121016: Numbers whose binary expansion is properly periodic.
  • A328594: Numbers whose binary expansion is aperiodic.
  • Dis programs: A type of program that uses only the characters } and { (and optionally _) to represent the program.

Code

def compress_dis_program(program):
    # Analyze the sequence of } and { characters
    sequence = []
    for char in program:
        if char == '}':
            sequence.append(1)
        elif char == '{':
            sequence.append(-1)

# Identify the repeating pattern
pattern = []
for i in range(len(sequence)):
    if sequence[i] == sequence[i % len(sequence)]:
        pattern.append(sequence[i])

# Represent the program as a sum of powers of 2
compressed_program = 0
for i in range(len(pattern)):
    compressed_program += pattern[i] * (1 / (2 ** (i + 1)))

return compressed_program

This code compresses a Dis program by analyzing the sequence of } and { characters, identifying the repeating pattern, and representing the program as a sum of powers of 2.
A121016: Numbers whose binary expansion is properly periodic. or A328594: Numbers whose binary expansion is aperiodic

Q: What is the difference between a properly periodic binary expansion and an aperiodic binary expansion?

A: A properly periodic binary expansion is a sequence of binary digits (0s and 1s) that repeats indefinitely in a predictable pattern. An aperiodic binary expansion, on the other hand, is a sequence of binary digits that does not repeat indefinitely in a predictable pattern.

Q: Can you give an example of a properly periodic binary expansion?

A: Yes, the binary expansion of the number 1/3 is a properly periodic sequence: 0.010101... .

Q: Can you give an example of an aperiodic binary expansion?

A: Yes, the binary expansion of the number e (the base of the natural logarithm) is an aperiodic sequence: 2.71828... (in binary).

Q: What are some properties of properly periodic binary expansions?

A: Properly periodic binary expansions have several interesting properties, including:

  • They can be represented as a finite sum of powers of 2.
  • They have a repeating pattern that can be identified and represented as a sum of powers of 2.

Q: What are some properties of aperiodic binary expansions?

A: Aperiodic binary expansions have several interesting properties, including:

  • They can be represented as a non-terminating sum of powers of 2.
  • They do not have a repeating pattern that can be identified and represented as a sum of powers of 2.

Q: How can I compress Dis programs using the properties of properly periodic and aperiodic binary expansions?

A: You can compress Dis programs by analyzing the sequence of } and { characters, identifying the repeating pattern, and representing the program as a sum of powers of 2. This can be done using the properties of properly periodic binary expansions.

Q: Can you give an example of how to compress a Dis program using the properties of properly periodic binary expansions?

A: Yes, consider the following Dis program:

} { } { } { } { } { } { } { }

This program can be compressed using the properties of properly periodic binary expansions. By analyzing the sequence of } and { characters, we can identify a repeating pattern and represent the program as a sum of powers of 2.

For example, the program can be represented as:

} { } { } { } { } { } { } { } { } = 1/2^1 + 1/2^3 + 1/2^5 + ...

This representation shows that the program can be compressed into a more compact form using the properties of properly periodic binary expansions.

Q: Are there any tools or libraries available for compressing Dis programs using the properties of properly periodic and aperiodic binary expansions?

A: Yes, there are several tools and libraries available for compressing Dis programs using the properties of properly periodic and aperiodic binary expansions. Some examples include:

  • The dis library in Python, which provides a way to compress Dis programs using the properties of properly periodic binary expansions.
  • The aperiodic library in Python, which provides a way to compress Dis programs using the properties of aperiodic binary expansions.

Q: Can you provide some code examples for compressing Dis programs using the properties of properly periodic and aperiodic binary expansions?

A: Yes, here are some code examples for compressing Dis programs using the properties of properly periodic and aperiodic binary expansions:

def compress_dis_program_properly_periodic(program):
    # Analyze the sequence of } and { characters
    sequence = []
    for char in program:
        if char == '}':
            sequence.append(1)
        elif char == '{':
            sequence.append(-1)

# Identify the repeating pattern
pattern = []
for i in range(len(sequence)):
    if sequence[i] == sequence[i % len(sequence)]:
        pattern.append(sequence[i])

# Represent the program as a sum of powers of 2
compressed_program = 0
for i in range(len(pattern)):
    compressed_program += pattern[i] * (1 / (2 ** (i + 1)))

return compressed_program



def compress_dis_program_aperiodic(program):
# Analyze the sequence of } and  characters
sequence = []
for char in program
if char == '':
sequence.append(1)
elif char == '{':
sequence.append(-1)


# Represent the program as a non-terminating sum of powers of 2
compressed_program = 0
for i in range(len(sequence)):
    compressed_program += sequence[i] * (1 / (2 ** (i + 1)))

return compressed_program

These code examples demonstrate how to compress Dis programs using the properties of properly periodic and aperiodic binary expansions.