In[]:=
Flatten[Transpose[With[{lr=LinearRecurrence[{3,-3,1},#,10]},1/Ratios[lr]]&/@{{1,10,28},{1,5,12},{2,7,15}}]]
Out[]=

1
10
,
1
5
,
2
7
,
5
14
,
5
12
,
7
15
,
28
55
,
6
11
,
15
26
,
55
91
,
22
35
,
13
20
,
91
136
,
35
51
,
40
57
,
68
95
,
51
70
,
57
77
,
190
253
,
35
46
,
77
100
,
253
325
,
92
117
,
50
63
,
325
406
,
117
145
,
126
155

In[]:=
FindSequenceFunction[%,u]
Out[]=
u+
2
u
12+7u+
2
u
In[]:=
Factor[%]
Out[]=
u(1+u)
(3+u)(4+u)
{{1,1},{2,1},{3,1},{4,1},{5,1},{6,1},{7,1},{8,1},{9,1},{10,1},{11,1},{12,1},{13,1},{14,1},{15,1},{16,1},{17,1},{18,1},{19,1},{20,1},{21,1}},4,
1
3
,5,
1
2
,6,
3
5
,7,
2
3
,8,
5
7
,9,
3
4
,10,
7
9
,11,
4
5
,12,
9
11
,13,
5
6
,14,
11
13
,15,
6
7
,16,
13
15
,17,
7
8
,18,
15
17
,19,
8
9
,20,
17
19
,21,
9
10
,22,
19
21
,5,
1
10
,6,
1
5
,7,
2
7
,8,
5
14
,9,
5
12
,10,
7
15
,11,
28
55
,12,
6
11
,13,
15
26
,14,
55
91
,15,
22
35
,16,
13
20
,17,
91
136
,18,
35
51
,19,
40
57
,20,
68
95
,21,
51
70
,22,
57
77
,23,
190
253
,6,
1
15
,7,
1
7
,8,
3
14
,9,
5
18
,10,
1
3
,11,
21
55
,12,
14
33
,13,
6
13
,14,
45
91
,15,
11
21
,16,
11
20
,17,
39
68
,18,
91
153
,19,
35
57
,20,
12
19
,21,
68
105
,22,
51
77
,23,
171
253
,24,
95
138
,8,
1
28
,9,
1
14
,10,
17
126
,11,
85
462
,12,
31
132
,13,
365
1287
,14,
327
1001
,15,
1097
3003
,16,
437
1092
,17,
668
1547
,18,
1973
4284
,19,
1415
2907
,20,
3959
7752
,21,
3613
6783
,22,
7279
13167
,23,
19227
33649
,24,
3127
5313
,25,
16057
26565

In[]:=
FindSequenceFunction[Last/@#,u]&/@{{1,1},{2,1},{3,1},{4,1},{5,1},{6,1},{7,1},{8,1},{9,1},{10,1},{11,1},{12,1},{13,1},{14,1},{15,1},{16,1},{17,1},{18,1},{19,1},{20,1},{21,1}},4,
1
3
,5,
1
2
,6,
3
5
,7,
2
3
,8,
5
7
,9,
3
4
,10,
7
9
,11,
4
5
,12,
9
11
,13,
5
6
,14,
11
13
,15,
6
7
,16,
13
15
,17,
7
8
,18,
15
17
,19,
8
9
,20,
17
19
,21,
9
10
,22,
19
21
,5,
1
10
,6,
1
5
,7,
2
7
,8,
5
14
,9,
5
12
,10,
7
15
,11,
28
55
,12,
6
11
,13,
15
26
,14,
55
91
,15,
22
35
,16,
13
20
,17,
91
136
,18,
35
51
,19,
40
57
,20,
68
95
,21,
51
70
,22,
57
77
,23,
190
253
,6,
1
15
,7,
1
7
,8,
3
14
,9,
5
18
,10,
1
3
,11,
21
55
,12,
14
33
,13,
6
13
,14,
45
91
,15,
11
21
,16,
11
20
,17,
39
68
,18,
91
153
,19,
35
57
,20,
12
19
,21,
68
105
,22,
51
77
,23,
171
253
,24,
95
138
,8,
1
28
,9,
1
14
,10,
17
126
,11,
85
462
,12,
31
132
,13,
365
1287
,14,
327
1001
,15,
1097
3003
,16,
437
1092
,17,
668
1547
,18,
1973
4284
,19,
1415
2907
,20,
3959
7752
,21,
3613
6783
,22,
7279
13167
,23,
19227
33649
,24,
3127
5313
,25,
16057
26565

Out[]=
1,
u
2+u
,
u+
2
u
12+7u+
2
u
,
u+
2
u
20+9u+
2
u
,
y.[2+n.]
[u]
MapIndexed[]
In[]:=
FindSequenceFunction[Last/@#,u]/.u->(s-(#[[1,1]]-1))&/@{{1,1},{2,1},{3,1},{4,1},{5,1},{6,1},{7,1},{8,1},{9,1},{10,1},{11,1},{12,1},{13,1},{14,1},{15,1},{16,1},{17,1},{18,1},{19,1},{20,1},{21,1}},4,
1
3
,5,
1
2
,6,
3
5
,7,
2
3
,8,
5
7
,9,
3
4
,10,
7
9
,11,
4
5
,12,
9
11
,13,
5
6
,14,
11
13
,15,
6
7
,16,
13
15
,17,
7
8
,18,
15
17
,19,
8
9
,20,
17
19
,21,
9
10
,22,
19
21
,5,
1
10
,6,
1
5
,7,
2
7
,8,
5
14
,9,
5
12
,10,
7
15
,11,
28
55
,12,
6
11
,13,
15
26
,14,
55
91
,15,
22
35
,16,
13
20
,17,
91
136
,18,
35
51
,19,
40
57
,20,
68
95
,21,
51
70
,22,
57
77
,23,
190
253
,6,
1
15
,7,
1
7
,8,
3
14
,9,
5
18
,10,
1
3
,11,
21
55
,12,
14
33
,13,
6
13
,14,
45
91
,15,
11
21
,16,
11
20
,17,
39
68
,18,
91
153
,19,
35
57
,20,
12
19
,21,
68
105
,22,
51
77
,23,
171
253
,24,
95
138
,8,
1
28
,9,
1
14
,10,
17
126
,11,
85
462
,12,
31
132
,13,
365
1287
,14,
327
1001
,15,
1097
3003
,16,
437
1092
,17,
668
1547
,18,
1973
4284
,19,
1415
2907
,20,
3959
7752
,21,
3613
6783
,22,
7279
13167
,23,
19227
33649
,24,
3127
5313
,25,
16057
26565

Out[]=
1,
-3+s
-1+s
,
-4+
2
(-4+s)
+s
12+7(-4+s)+
2
(-4+s)
,
-5+
2
(-5+s)
+s
20+9(-5+s)+
2
(-5+s)
,
y.[2+n.]
[-7+s]
In[]:=
Simplify[%]
Out[]=
1,
-3+s
-1+s
,
12-7s+
2
s
(-1+s)s
,
20-9s+
2
s
(-1+s)s
,
y.[2+n.]
[-7+s]
In[]:=
Factor[%]
Out[]=
1,
-3+s
-1+s
,
(-4+s)(-3+s)
(-1+s)s
,
(-5+s)(-4+s)
(-1+s)s
,
y.[2+n.]
[-7+s]
In[]:=
FunctionExpand
y.[2+n.]
[-7+s]
Out[]=
y.[2+n.]
[-7+s]
In[]:=
Table(s(s-1))
y.[2+n.]
[-7+s],{s,10}
Out[]=
0,2
y.[2+n.]
[-5],6
y.[2+n.]
[-4],12
y.[2+n.]
[-3],20
y.[2+n.]
[-2],
43955104974951215464938760
1990983621047655999459447
,
11591549096363
2965138749081
,2,
36
7
,
85
7
