Thread: Difference tables View Single Post
 I wrote this whole thing out in Notepad before logging in this time. That ought to prevent the earlier mishap I tested $x^2 3^x + x 4^x$ see how far my ideas for determining the function for a sequence could go. As I knew the answer a priori, this was a test to see if I could identify the answer in the tables. To start with, I wrote the sequence with its difference table. I figured eight terms should be enough to find the pattern: $$\begin{array}{ccc ccc ccc ccc ccc} 0 && 7 && 68 && 435 && 2320 && 11195 && 50820 && 221851 \\ & 7 && 61 && 367 && 1885 && 8875 && 39625 && 171031 \\ && 54 && 306 && 1518 && 6990 && 30750 && 131406 \\ &&& 252 && 1212 && 5472 && 23760 && 100656 \\ &&&& 960 && 4260 && 18288 && 76896 \\ &&&&& 3300 && 14028 && 58608 \\ &&&&&& 10728 && 44580 \\ &&&&&&& 33852 \end{array}$$ There is no obvious pattern that I could see in the \ oriented diagonals, so I made a new difference table, taking the first diagonal of this table to be the first row of the new table: $$\begin{array}{ccc ccc ccc ccc ccc} 0 && 7 && 54 && 252 && 960 && 3300 && 10728 && 33852 \\ & 7 && 47 && 198 && 708 && 2340 && 7428 && 23124 \\ && 40 && 151 && 510 && 1632 && 5088 && 15696 \\ &&& 111 && 359 && 1122 && 3456 && 10608 \\ &&&& 248 && 763 && 2334 && 7152 \\ &&&&& 515 && 1571 && 4818 \\ &&&&&& 1056 && 3247 \\ &&&&&&& 2191 \end{array}$$ Nothing seems to be standing out, so I repeated the process: $$\begin{array}{ccc ccc ccc ccc ccc} 0 && 7 && 40 && 111 && 248 && 515 && 1056 && 2191 \\ & 7 && 33 && 71 && 137 && 267 && 541 && 1135 \\ && 26 && 38 && 66 && 130 && 274 && 594 \\ &&& 12 && 28 && 64 && 144 && 320 \\ &&&& 16 && 36 && 80 && 176 \\ &&&&& 20 && 44 && 96 \\ &&&&&& 24 && 52 \\ &&&&&&& 28 \end{array}$$ The first diagonal is starting to have spots where it decreases, which is behavior that hasn't been seen so far. I started the next difference table, but stopped at the third row, since an infinite sequence of zeros appeared at that point: $$\begin{array}{ccc ccc ccc ccc ccc} 0 && 7 && 26 && 12 && 16 && 20 && 24 && 28 \\ & 7 && 19 && -14 && 4 && 4 && 4 && 4 \\ && 12 && -33 && 18 && 0 && 0 && 0 \end{array}$$ If the row was entirely zeros, the first few numbers in the top row would have to change. The amount each of those numbers must change is important: 0-0, 7-3, 26-18, 12-0, 16-0, ..., and the change is 0, 3, 18. The first diagonal would then be 0, 4. These finite sequences are the key to the polynomials that multiply the exponentials. Namely, since 0, 3, 18 comes from the diagonal of the third difference table, it will contribute: $$3^x \left( 0 \times \frac{ \binom{x}{0} }{3^0} + 3 \times \frac{ \binom{x}{1} }{3^1} + 18 \times \frac{ \binom{x}{2} }{3^2} \right) = x^2 3^x$$ Likewise, since 0, 4 comes from the diagonal of the fourth difference table, it will contribute: $$4^x \left( 0 \times \frac{ \binom{x}{0} }{4^0} + 4 \times \frac{ \binom{x}{1} }{4^1} \right) = x 4^x$$ Thus, the whole function is revealed: $x^2 3^x + x 4^x$. A table of the first diagonals can be an abbreviated way of expressing much of the above difference tables. Although I did not complete the last difference table in this description, I include the complet first diagonal here. It alternates, indicating that a polynomial was present at some stage, as we already know: $$\begin{array}{cccc cccc} 0 & 7 & 68 & 435 & 2320 & 11195 & 50820 & 221851 \\ 0 & 7 & 54 & 252 & 960 & 3300 & 10728 & 33852 \\ 0 & 7 & 40 & 111 & 248 & 515 & 1056 & 2191 \\ 0 & 7 & 26 & 12 & 16 & 20 & 24 & 28 \\ 0 & 7 & 12 & -45 & 96 & -165 & 252 & -357 \end{array}$$