$ \sqrt{2\pi} $（$ 2\pi $の平方根）の小数以下の数列で幾つかの数列パターンの、出現回数、出現箇所などを、探索してみた。
まずは、小数1000桁まで。
$ \sqrt{2\pi} $ = 2.506628274631000...2170058177
000: 2 @ 13 269
111: 0
222: 0
333: 0
444: 0
555: 1 @ 834
666: 1 @ 315
777: 2 @ 338 749
888: 0
999: 1 @ 433
123: 1 @ 169
321: 0
0, 1, ..., 9の出現回数
0: 98
1: 100
2: 93
3: 104
4: 97
5: 102
6: 114
7: 89
8: 116
9: 87
探索範囲を100万桁に拡げ、パターン長さを6に伸ばす。
$ \sqrt{2\pi} $ = 2.506628274631000...4590097547
000000: 0
111111: 1 @ 225974
222222: 0
333333: 1 @ 658372
444444: 0
555555: 3 @ 123424 591298 834545
666666: 0
777777: 3 @ 187121 510916 765881
888888: 0
999999: 0
123456: 0
654321: 1 @ 407447
0, 1, ..., 9, 000, 111, ..., 999の出現回数
0: 100542
1: 99589
2: 99962
3: 100118
4: 99638
5: 100409
6: 100134
7: 100029
8: 99839
9: 99740
000: 951
111: 875
222: 882
333: 899
444: 863
555: 892
666: 896
777: 915
888: 922
999: 901
探索範囲を1億桁に拡げ、パターン長さを8に伸ばす。
$ \sqrt{2\pi} $ = 2.506628274631000...7235781668
00000000: 1 @ 57020545
11111111: 2 @ 81224521 89904572
22222222: 1 @ 60366485
33333333: 2 @ 34851906 42860589
44444444: 1 @ 25163312
55555555: 3 @ 7562426 32170242 91203809
66666666: 1 @ 20252394
77777777: 2 @ 22681143 72447997
88888888: 1 @ 71778819
99999999: 1 @ 2200922
12345678: 0
87654321: 2 @ 10869550 37169954
0, 1, ..., 9, 000, 111, ..., 999の出現回数
0: 9999526
1: 10000374
2: 9999518
3: 9998511
4: 9995541
5: 100002192
6: 100005513
7: 9997078
8: 100001177
9: 100000570
000: 91084
111: 90697
222: 90869
333: 91003
444: 90866
555: 90663
666: 90605
777: 90630
888: 91309
999: 91777