Pell series are progressions approximating geometry of the octagram. It's successive factors are approximating a constant θ = 2.414 derived from the octagram. The further we follow the progression the more accurate is the approximation.
The next factor of Pell series is obtained if a factor is multiplied by 2 and preceding factor added, e.g. 2x2+1 = 5, 2x5+2 = 12, 2x12+5 = 29, ...
TABLE OF PELL SERIES
1
1
2
5
12
29
70
169
408
985
...
2
1
3
7
17
41
99
239
577
1393
...
3
1
4
9
22
53
128
309
746
1801
...
4
1
5
11
27
65
157
379
915
2209
...
5
1
6
13
32
77
186
449
1084
2617
...
6
1
7
15
37
89
215
519
1253
3025
...
7
1
8
17
42
101
244
589
1422
3433
...
8
1
9
19
47
113
273
659
1591
3841
...
9
1
10
21
52
125
302
729
1760
4249
...
10
1
11
23
57
137
331
799
1929
4657
...
11
1
12
25
62
149
360
869
2098
5065
...
12
1
13
27
67
161
389
939
2267
5473
...
13
1
14
29
72
173
418
1009
2436
5881
...
14
1
15
31
77
185
447
1079
2605
6289
...
15
1
16
33
82
197
476
1149
2774
6697
...
16
1
17
35
87
209
505
1219
2943
7105
...
17
1
18
37
92
221
534
1289
3112
7513
...
18
1
19
39
97
233
563
1359
3281
7921
...
19
1
20
41
102
245
592
1429
3450
8329
...
20
1
21
43
107
257
621
1499
3619
8737
...
21
1
22
45
112
269
650
1569
3788
9145
...
22
1
23
47
117
281
679
1639
3957
9553
...
23
1
24
49
122
293
708
1709
4126
9961
...
24
1
25
51
127
305
737
1779
4295
10369
...
25
1
26
53
132
317
766
1849
4464
10777
...
26
1
27
55
137
329
795
1919
4633
11185
...
27
1
28
57
142
341
824
1989
4802
11593
...
28
1
29
59
147
353
853
2059
4971
12001
...
29
1
30
61
152
365
882
2129
5140
12409
...
30
1
31
63
157
377
911
2199
5309
12817
...
Numbers derived from these compositions are widely used in art and architecture as ratios of the composition. An octagram is a pair of rotated squares, and therefore the progressions also contain the most rudimentary relation between a side of a square and its diagonal - √2. The approximation of √2 is concealed in the difference of successive factors, for example if we take the first progression 1 2 5 12 29 70 169 ... a
diagonal of a square with base 29 is 70-29 = 41, and again the further the progression grows the more accurate approximation we get.
