Can the result of a change of one simple rule to build a Pascal's triangle be mathematically explained?

Question

The only rule that is changed is this: when adding two numbers from row $(n-1)$ to get the number $N_n$ of row $n$ we also add the number from row $(n-2)$ that is aligned (vertically) with the position of the number $N_n$.

We start with the same triangle $$1$$ $$1---1$$ The next line will be $$1---3---1$$ because we added the first $1$ aligned with the result of adding $1+1$ from row $2$. If we keep using this rule, we will get a Pascal's triangle whose diagonals are very interesting. Since I don't know how to format a Pascal's triangle with mathjax, I will simply list few diagonals.

The first diagonal is: $$1--3--5--7--9--11--13--15--15--17--19--21...$$

The second diagonal is given by $A001844$ which gives centered square numbers. $$1--5--13--25--41--61--85--113--145--181--221...$$ The third diagonal is given by $A001845$ which gives centered octahedral numbers (also called crystal ball sequence for cubic lattice).
$$1--7--25--63--129--231--377--575--833--1159...$$

The fourth diagonal is given by $A001846$ which gives centered 4-dimensional orthoplex numbers (also called crystal ball sequence for 4-dimensional cubic lattice).
$$1--9--41--129--321--681--1289--2241--3649...$$

The fifth diagonal is given by $A001847$ which gives crystal ball sequence for 5-dimensional cubic lattice numbers. The sixth diagonal is given by $A001848$ which gives crystal ball sequence for 6-dimensional cubic lattice numbers. I suppose the next sequence will give the 7-dimensional cubic lattice numbers ( I checked it ) and higher.

Some of the formulas to generate the numbers are given in the OEIS sequences given above.

The first $7$ rows of the this triangle look like this: $$1$$ $$1---1$$ $$1---3---1$$ $$1---5---5---1$$ $$1---7---13---7---1$$ $$1---9---25---25---9---1$$ $$1---11---41---63---41---11---1$$

Note the pattern that repeats indefinitely formed by multiplying numbers under the starting $1$ at the top in the following way: $$1*3+1*1=4=2^2=(1+1)^2$$ $$3*13+5*5=64=8^2=(3+5)^2$$ $$13*63+25*25=1444=38^2=(13+25)^2$$.

The triangle has other properties that deserve to be mentioned. If we add term by term the first diagonal and the second to get: $$(1+1),(3+5), (5+13), (7+25), (9+41), (11+61), (13+85)...$$ we get the sequence $2n^2$.

If we add the second diagonal and the third term by term, we get the sequence A035597 which gives the number of points of L1 norm 3 in cubic lattice Z^n. Its formula is ($4n^3+2n)/3$:

$$(1+1), (5+7), (13+25), (25+63), (41+129), (61+231)...$$

But we can also get new numbers by multiplying term by term the first and second diagonals. We get OEIS A005917 Rhombic dodecahedral numbers: $a(n) = n^4 - (n - 1)^4$
$$1, 15, 65, 175, 369, 671, 1105, 1695, 2465, 3439...$$

There are probably more hidden patterns waiting to be found in this triangle. Only a systematic search can find them.

There are many questions that come to mind.

1-How come one simple modification of a rule provides such a change.
2-What mathematics (formulas, theorems...) is common to both triangles (assuming some features are common to both triangles which is not obvious at all at this point).
3-Have the effect of other modifications of the rule to build a Pascal's triangle been systematically studied before? ( For example, one can think of a classical Pascal's triangle where the result is squared...).

If someone can think of more meaningful tags, please add them or change them.

Answer 1

a) In dealing with such triangular arrays, it is convenient to arrange them as a Lower Triangular array (matrix), indexed from $0$. That greatly simplifies notation, and allows matrix "tools" to be applied.

b) In this scheme, the LT Pascal matrix ($\bf P$) is defined by the recurrence $$ \left\{ \matrix{ p_{\,0,\,m} = \left[ {0 = m} \right] \hfill \cr p_{\,n,\,m} = p_{\,n - 1,\,m} + p_{\,n - 1,\,m - 1} \hfill \cr} \right. $$ where $[P]$ denotes the Iverson bracket,
or more compactly as $$ p_{\,n,\,m} = \left[ {0 = n} \right]\left[ {0 = m} \right] + \left[ {1 \le n} \right]\left( {p_{\,n - 1,\,m} + p_{\,n - 1,\,m - 1} } \right) $$ Note that the Initial Conditions are as much qualifying as the recurrence itself.

c) The LT "modified" Pascal matrix ($\bf Q$) you propose will read $$ q_{\,n,\,m} = \left[ {0 = n} \right]\left[ {0 = m} \right] + \left[ {1 \le n} \right]\left( {q_{\,n - 1,\,m} + q_{\,n - 1,\,m - 1} } \right) + \left[ {2 \le n} \right]q_{\,n - 2,\,m} $$ Note that it is a second degree recurrence, instead of a first degree, and that it involves three precursors instead of two.
No doubt that it will produce quite a different result, and indeed much interesting, and here I will limit to
summarily describe some, in addition to those already indicated.

The matrix ($0 \ldots 10 \times 0 \ldots 10$) is

d) The double o.g.f. will be $$ G(x,y) = \sum\limits_{0\, \le \,n} {\sum\limits_{0\, \le \,m} {q_{\,n,\,m} x^{\,n} y^{\,m} } } = {1 \over {1 - x\left( {1 + y} \right) - x^{\,2} }} $$ which for $m=0 \to y=0$ is that of the Fibonacci Numbers, which are in fact in the first column.
The following columns are convolutions of the Fibonacci N.

Putting $y=1$, we get the o.g.f. of the row sums, which corresponds to that of the Pell Numbers (shifted by one).

e) In relation to the Pascal matrix $\bf P$, it turns out that both are similar to the bidiagonal matrix $\bf I + \bf E$, and thus are similar to each other. $$ {\bf I + E} = \left( {\matrix{ 1 & 0 & 0 & \cdots \cr 1 & 1 & 0 & \cdots \cr 0 & 1 & 1 & \cdots \cr \vdots & \ddots & \ddots & \ddots \cr } } \right) $$

Moreover $$ {\bf Q}\,{\bf P}^{\, - \,{\bf 1}} = {\bf A}\quad \left| {\;a_{\,n,\,m} :\left[ {0 = \left( {n + m} \right)\bmod 2} \right]\left( \matrix{ {{n + m} \over 2} \cr m \cr} \right)} \right. $$

.. and much else.

Answer 2

This is the doorway to whole world of Number-Theory, Combinatorics pleasure !

With regards to your third question, a favourite of mine is the Trinomial Coefficients, where each term is the sum of the three above,

$$1$$ $$1, 1, 1$$ $$1, 2, 3, 2, 1$$ $$1, 3, 6, 7, 6, 3, 1$$ $$1, 4, 10, 16, 19, 16, 10, 4, 1$$

This expands $(1+x+x^2)^n$ in much the same way as Pascal's Triangle expands the Binomial's $(1+x)^n$

So, for example, that last row tells us that, $$(1+x+x^2)^4$$ $$=1+4x+10x^2+16x^3+19x^4+16x^5+10x^6+4x^7+x^8$$

There are other Multinomials, of course, of higher orders.

I'm looking forward to the other answers that your interesting question will generate.