I'm working on a mathematical model which requires me to generalize the elements of an infinite set. The element n is the nth finite sequence of the set.
For:
n = 1: {0}
n = 2: {0, 1, 3, 5, 6, 7, 9, 15}
n = 3: {0, 1, 2, 5, 26, 29, 32, 81, 83, 87, 107, 112, 113, 116, 135, 140, 141, 142, 143, 161, 162, 194, 224, 351, 353, 356, 364, 365, 377, 647, 728, 1514, 1536, 1538, 1595, 1601, 1617, 1619, 1862, 2271, 2273, 2300, 2460, 2462, 2541, 2543, 2561, 3028, 3029, 3168, 3280, 3281, 3289, 3293, 3785, 3806, 3968, 4001, 4009, 4017, 4018, 4019, 4022, 4037, 4046, 4048, 4049, 4069, 4100, 4130, 4289, 4373, 5975, 6056, 6209, 6317, 6336, 6479, 6560, 6673, 6674, 7381, 8180, 9104, 9113, 9833, 9840, 9841, 9842, 9854, 10598, 10609, 10610, 11453, 12301, 13121, 13265, 13346, 14001, 14741, 15227, 15665, 16402, 17141, 17222, 17411, 18146, 18914, 19115, 19331, 19520, 19601, 19682}
OEIS doesn't have the sequence n = 2 or n = 3, let alone a master sequence, so I'm kinda stuck. Does anyone know any math magic to help me generalize the set of sequences. I can provide sequences for n > 3 if necessary. I can also provide the algorithm (c-code) and/or system of equations I'm using to generate these finite sequences.
Weird, right? I have an indirect method for generating these sequences, but no direct function. If you don't have an answer, that's understandable, but any advice for a plan of attack would surely help.
Thanks for your time, assuming you take the time to read this.
Update as requested:
Origin of the family of sequences:
Generate an ordered, finite set $T_n: T_n = \{A_1,...,A_\mu...,A_n\} $, and an operator $\circ: A_p \circ A_q ∈ T_n$.
For $A_\mu ∈ T_n,\ P(A_\mu)$ is the position of $A_\mu$ in set $T_n$.
Further, let:
$1. \ \ \ Q_k := (A_X \circ A_Y)_k$
$2.\ \ \ P(Q_k) = \left \lfloor{k\cdot n^{n*P(A_X)+P(A_Y)+1-{n^2}}} \right \rfloor\mod n:0\le k\lt n^{n^2}$
Then associativity of $\circ$ on the set $T_n$ requires that:
$ \bigl((A_X \circ A_Y)_k \circ A_Z\bigr)_k = \bigl(A_X \circ (A_Y \circ A_Z)_k\bigr)_k$.
Once you select a given $n$, we seek the instances $k$ for which the set $T_n$ is associative. There is typically more than one solution, $k$. The solutions of k for a given n generate the sequences written above.
Hence my dilemma:
I have an algorithm for generating the numbers, but can't see through the math in order to find a general function that generates all of the finite sequences, and I'm hoping for a fresh perspective, maybe a technique or something, to hopefully find said function.
Again, thanks for your input. I can do the work, but I'll appreciate a nudge in the right direction. :)