All About Golomb Sequence

The Golomb Sequence is named after Solomon w. Golomb. Solomon Wolf Golomb was an American mathematician and engineer.

He worked as a professor at the University of Southern California. He was best known for his mathematical games. The sequence is also called Silverman’s Sequence.

Introduction Golomb Sequence

Golomb Sequence can be defined as: –

The Sequence is a non-decreasing (no element is less than the element before; in simple words, we can say that every element is greater than or equal to the one before it) integer sequence where the n-th term is equal to several times n appears in the sequence.

The first few values of the Golomb’s Sequence are: –
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, ……

Let us know the explanation of these terms: –

• The first term is 1, and one appears 1 time.
• The second term is 2, and two appear 2 times.
• The third term is 2, and three appears 2 times.
• The fourth term is 3, and four appears 3 times
• And so on………………………………………………………

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Examples on Golomb Sequence

Now, we will solve questions related to the Golomb’s sequence: –

Example 1:

A positive integer n is given to you. Do you have to find the first n terms of the Golomb’s sequence?

Case 1 = If n is 3, which means
Input = 3
The output will be, or the first n terms of the sequence will be 1,2,2.

Case 2 = If n is 7, which means
Input = 7
The output will be, or the first n terms of the sequence will be 1,2,2,3,3,4,4,4,5,5,5.

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Example 2:

If the Sixth term of a Golomb sequence is 4, how many times will 6 appear in the sequence?

Write the Sequence first: – 1,2,2,3,3,4,4,4,5,5,5…………….
If the sixth term is 4, six will appear four times in the sequence.

Golomb Sequence: – 1,2,2,3,3,4,4,4,5,5,5,6,6,6,6………………….

Practice

1. A positive integer n = 6 is given to you; what will be the first n terms of the Golomb sequence?
2. If the fourth term of a Golomb’s sequence is 3, how many times will 4 appear in the Golomb sequence?