Triangle of Fibonacci

Question

I was reading in a book a presentation about the arithmetic triangle of Fibonacci (to me it also looks like the pascal triangle).
The figure presented is as follows:

The text says:
Having arranged the consecutive odd integers in a triangular array as in the figure above, Fibonacci observed that the $k$ terms of the $k$-th row form an arithmetic progression of mean value $k^2$ hence the sum of the terms of the $k$-th row is $k \cdot k^2$ or $k^3$ and the sum of all terms in $n$ consecutive rows is $1^3 + 2^3 + 3^3 + ... + n^3$

1. To begin with I don't understand where is the arrangement of consecutive odd integers in the array. I can see $1, 3, 5, 7$ in the array but is there some special arrangement in the array I am not noticing?
2. My understanding is that an arithmetic progression is just another name for an arithmetic sequence. And the mean value of a sequence is the middle number if the number of elements in the sequence is odd or the average of the $2$ elements in the middle e.g. for $1, 3, 5, 7, 9$ the mean is $5$ while for $1, 3, 5, 7, 9, 11$ the mean is $\frac{5 + 7}{2}$. Based on this I don't understand the statement "the $k$ terms of the $k$-th row form an arithmetic progression of mean value $k^2$". Each row from top to bottom has a half arithmetic progression because it increases till the middle and then decreases so I am not sure if that counts as an arithmetic progression. More so I can't find the value $k^2$ for any row. E.g. for $k = 1$ we have ${1}$ so the mean is $1^2$. For $k =3$ we have ${1,2,1}$ and the mean value is $2 \mathrel{{=}\llap{/\,}} 3^2 $

Is there any typo in the text or else what am I misunderstanding here?

Answer 1

The second row should have $3,5$; the third row $7,9,11$; the fourth row $13,15,17,19$ and so son.