I´m trying to find a sequence function for the following sequence:
$0, 1, 85, 419, 973, 1747, 2741, 3955, 5389, 7043, 8917, 11011, 13325, 15859, 18613, 21587, 24781, 28195$
The first term is generated from a parameter $n=1$, the second from $n=2$, and so on.
When using the Wolfram´s function FindSequenceFunction[], it doesn´t provide any useful result. However, if I remove the first 2 terms $0,1$, it returns the sequence function $f(n)=110 n^2+4 n-29$. Since it misses the first two terms, I evaluate $f(n-2)$ to get the actual corrected result, which equals to $110 n^2-436 n+403$.
The problem is that the resulting $f(n-2)$ function equals $77$ instead of $0$ when $n=1$, and when $n=2$, it returns $-29$ instead of $1$. I wonder if there exists any transformation I can apply to the $f(n-2)$ function so that it returns the correct result for $n=1$ and $n=2$, considering all the current outputs for $n>2$ as correct.
Also, I tried similar sequences (created from the 2. case of this question, which counts arrangements of clusters with $k$ kings on a $n\times n$ chessboard), for example:
The sequence from this question counts $4$ king clusters, but for $k<4$:
$(k=2)=>0, 6, 20, 42, 72, 110, 156, 210, 272$
$(k=3)=>0, 4, 48, 132, 256, 420, 624, 868, 1152, 1476$
Returning the correct generating functions:
$(k=2)=>4 n^2-6 n+2$
$(k=3)=>20 n^2-56 n+36$
As you can see, for 2 and 3 values of $k$ Wolfram finds a sequence function with the initial values $0,6$ and $0,4$ in it. So, I also wonder if there exists a way of proving if a sequence function for higher values of $k$ can exist or not depending on the result Wolfram returns when using the first $0,\cdots$ initial values inside FindSequenceFunction[].