Sequences that name there own position
GENERAL CONSIDERATIONS / EXPECTED VALUES
It can be very interesting to explore the self referential qualities of a digit sequence. One special kind of that is a correspondence between a subsequence and its absolute position in the whole sequence.
If we find the sequence "klm...nop" exactly at position (klm...nop) we call this a CUCKOO POSITION (German: Kuckuckszahl).
Another definition of CUCKOO could be:
SEQUENCE = POSITION of SEQUENCE
The position predicts the following sequence like the couckoo bird shouts its own name.
The first Cuckoo Position K1 is not hard to find because Pi starts with "1" at position 1 (after the decimal point): 14159265...
But where comes the next Couckoo Position and how many of them will be found amongst our 100 billion digits?
RESULTS: All Cuckoo-Position up to 100.000.000.000
Verified 2017/11/03 when analyzing 1012 digits:
K9 = 656.430.109.694
RESULTS: Hits per Range
We can extend our view to NEAR CUCKOO POSITIONS which are not exact hits but pretty close to it.
This unfolds a basic statistical behaviour:
If you increase the data tenfold, you generally get one more cuckoo hit.
S = Sequences proofed; L = Length of Sequences
H = Hits (where ABS(Pos.-Value)<10)