was reading an article about general formula for $n^{th}$ term of a cubic sequence, and it mentioned the formula: $$u_n=an^3+bn^2+cn+d$$ Here, the values of $a, b, c$ & $d$ are calculated using the equations: $$6a=u_4-3u_3+3u_2-u_1,$$ $$12a+2b=u_3-2u_2+u_1,$$ $$7a+3b+c=u_2-u_1,$$ $$a+b+c+d=u_1$$
But there's no mention of the derivation of these equations calculating the values of $a, b, c, d.$
Now, such equations exist for linear, quadratic, cubic... so on for all higher degree sequences, but how are these equations derived?
I tried looking for derivation on YouTube, polynomial interpolation notes, numerical analysis books, but can't find any literature related to this.
Any book that covers this topic in detail?
Plainly take the definitions of $u_1,u_2,u_3,u_4$ by replacing $n$ with $1,2,3,4$ resp. :
$$\begin{cases} u_1&=&a+b+c+d&(a)\\ u_2&=&8a+4b+2c+d&(b)\\ u_3&=&27a+9b+3c+d&(c)\\ u_4&=&64a+16b+4c+d&(d) \end{cases}\tag{1}$$
Then, take (a) as it is, subtract (a) to (b) in order to obtain your second equation, then make the combination $(a)+(c)-2 \times (b)$ in order to get your third equation, etc.
Remark 1 : Have you noticed that the coefficients in the RHS of your equations are :
$$\begin{array}{rrrr}1&-3&3&-1\\ &1&-2&1\\ &&1&-1\\ &&&1\end{array}\tag{2}$$
which are the binomial coefficients obtained in the development of $(a-b)^n$ ?
There is an explanation that necessitates to turn to matrix formulation.
This is connected to the so-called LU decomposition (L = lower triangular matrix, U = upper triangular matrix) of the coefficient matrix of system (1) above under the following reversed form :
$$\underbrace{\begin{pmatrix}1&1&1&1\\ 1&2&4&8\\ 1&3&9&27\\ 1&4&16&64 \end{pmatrix}}_V=\underbrace{\begin{pmatrix}1&&&\\ 1&1&&\\ 1&2&1&\\ 1&3&3&1 \end{pmatrix}}_P\underbrace{\begin{pmatrix}1&1&1&1\\ &1&3&7\\ &&2&12\\ &&&6\\ \end{pmatrix}}_Q$$
where $P$ is a so-called Pascal matrix, $V$ a "Vandermonde" matrix, $Q$ being the matrix of coefficients you were given...
Remark 2 : a different explanation for the presence of coefficients as given by (2), in direct connection with the reference you give, is that they are equivalent to discrete derivatives of order 3,2,1,0, resp. For example, the discrete "order 3" derivation of the 3rd degree polynomial $an^3+...$ with respect to variable $n$ is a constant equal to $(an^3)'''=6a$, the LHS of your first equation.