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,,5,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,5,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,24,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,24,,25,
1
3
1
2
3
5
2
3
5
7
3
4
7
9
4
5
9
11
5
6
11
13
6
7
13
15
7
8
15
17
8
9
17
19
9
10
19
21
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
1
15
1
7
3
14
5
18
1
3
21
55
14
33
6
13
45
91
11
21
11
20
39
68
91
153
35
57
12
19
68
105
51
77
171
253
95
138
1
28
1
14
17
126
85
462
31
132
365
1287
327
1001
1097
3003
437
1092
668
1547
1973
4284
1415
2907
3959
7752
3613
6783
7279
13167
19227
33649
3127
5313
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,,5,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,5,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,24,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,24,,25,
1
3
1
2
3
5
2
3
5
7
3
4
7
9
4
5
9
11
5
6
11
13
6
7
13
15
7
8
15
17
8
9
17
19
9
10
19
21
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
1
15
1
7
3
14
5
18
1
3
21
55
14
33
6
13
45
91
11
21
11
20
39
68
91
153
35
57
12
19
68
105
51
77
171
253
95
138
1
28
1
14
17
126
85
462
31
132
365
1287
327
1001
1097
3003
437
1092
668
1547
1973
4284
1415
2907
3959
7752
3613
6783
7279
13167
19227
33649
3127
5313
16057
26565
Out[]=
1,,,,[u]
u
2+u
u+
2
u
12+7u+
2
u
u+
2
u
20+9u+
2
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,,5,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,5,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,24,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,,19,,20,,21,,22,,23,,24,,25,
1
3
1
2
3
5
2
3
5
7
3
4
7
9
4
5
9
11
5
6
11
13
6
7
13
15
7
8
15
17
8
9
17
19
9
10
19
21
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
1
15
1
7
3
14
5
18
1
3
21
55
14
33
6
13
45
91
11
21
11
20
39
68
91
153
35
57
12
19
68
105
51
77
171
253
95
138
1
28
1
14
17
126
85
462
31
132
365
1287
327
1001
1097
3003
437
1092
668
1547
1973
4284
1415
2907
3959
7752
3613
6783
7279
13167
19227
33649
3127
5313
16057
26565
Out[]=
1,,,,[-7+s]
-3+s
-1+s
-4++s
2
(-4+s)
12+7(-4+s)+
2
(-4+s)
-5++s
2
(-5+s)
20+9(-5+s)+
2
(-5+s)
In[]:=
Simplify[%]
Out[]=
1,,,,[-7+s]
-3+s
-1+s
12-7s+
2
s
(-1+s)s
20-9s+
2
s
(-1+s)s
In[]:=
Factor[%]
Out[]=
1,,,,[-7+s]
-3+s
-1+s
(-4+s)(-3+s)
(-1+s)s
(-5+s)(-4+s)
(-1+s)s
In[]:=
FunctionExpand[-7+s]
Out[]=
In[]:=
Table(s(s-1))[-7+s],{s,10}
Out[]=
0,2[-5],6[-4],12[-3],20[-2],,,2,,
43955104974951215464938760
1990983621047655999459447
11591549096363
2965138749081
36
7
85
7