The structure of each row

Fig 5.4 is a part of Fig 5.3.

There are four rows that are colored in red , blue , green and yellow respectively. We study these rows one by one.

Fig 5.5 present the case of
, and this number
is in the rectangle of gray color. If we start with the last number
and go leftward picking up every other number, then we have
which is an arithmetic sequence with common difference of
.

On the other hand if we start with
that is the second number from the right end of the row and go leftward picking up every other number, then we have
which starts as an arithmetic sequence with common difference of
and later it becomes an arithmetic sequence with common difference of
.

In this way numbers in Fig 5.5 can be seen as a combination of two different sequences, and we can see this as a decreasing sequence;

.

Here we present a prediction for the row when
. Fig 5.6 present the case that is a generalization of the case of
in Fig 5.5.

On the other hand if we start with that is the second number from the right end of the row and go leftward picking up every other number, then we have an arithmetic sequence with common difference of whose length is , where is a ceiling function, and hence the last number of this sequence is . This number is the first number of an arithmetic sequence with common difference of and the last number of this sequence is .

In this way numbers in Fig 5.6 can be seen as a combination of two different sequences, and we can see this as a decreasing sequence;

.

Similarly Fig 5.7 present the case of , and this number is in the rectangle of gray color. If we start with the last number and go leftward picking up every other number, then we have which is an arithmetic sequence with common difference of .

On the other hand if we start with that is the second number from the right end of the row and go leftward picking up every other number, then we have , which starts as an arithmetic sequence with common difference of and later it becomes an arithmetic sequence with common difference of .

In this way numbers in Fig 5.7 can be seen as a combination of two different sequences, and we can see this as a decreasing sequence;

. Here we present a prediction for the row when . Fig 5.8 present the case that is a generalization of the case of in Fig 5.7.

On the other hand if we start with that is the second number from the right end of the row and go leftward picking up every other number, then we have an arithmetic sequence with common difference of whose length is , and hence the last number of this sequence is . This number is the first number of an arithmetic sequence with common difference of and the last number of this sequence is .

In this way numbers in Fig 5.8 can be seen as a combination of two different sequences, and we can see this as a decreasing sequence;

.

On the other hand if we start with that is the second number from the right end of the row and go leftward picking up every other number, then we have , which starts as an arithmetic sequence with common difference of and the last number is 7, and the next number is 9. Note that the difference between and is . After that it becomes an arithmetic sequence with common difference of .

In this way numbers in Fig 5.9 can be seen as a combination of two different sequences, and we can see this as a decreasing sequence;

.

Here we present a prediction for the row when . Fig 5.10 present the case that is a generalization of the case of in Fig 5.9.

On the other hand if we start with that is the second number from the right end of the row and go leftward picking up every other number, then we have an arithmetic sequence with common difference of whose length is , and hence the last number of this sequence is . The next number is that is the first number of an arithmetic sequence with common difference of and the last number of this sequence is .

Note that the difference of and is 2.

Numbers in Fig 5.10 can be seen as a combination of two different sequences, and we can see this as a decreasing sequence;

.

On the other hand if we start with that is the second number from the right end of the row and go leftward picking up every other number, then we have an arithmetic sequence with common difference of whose length is , and hence the last number of this sequence is . The next number is that is the first number of an arithmetic sequence with common difference of and the last number of this sequence is .

Note that the difference of and is 2.

Numbers in Fig 5.10 can be seen as a combination of two different sequences, and we can see this as a decreasing sequence;

.