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General
| U0 = a, U1 = b and Un+1 = F*Un - Un-1 + c |
a, b and c are integers, F is a natural number.
If c = 0, we'll represent the series as U(a, b)F and call it a 'zero series'.
Unless indicated otherwise, 'series' will mean a series of the above type.
It all started unintentionally when I invented Havannah and noticed that the number of cells of the board, 169 at the time, was also a square. The question which would be the next centered hexagon that also was a square therewith presented itself. And how would squares and triangles intersect? And centered hexagons and triangles?
Obvious questions that resulted in second degree diophantine equations whose solutions turned out to involve recurrent series of the above type.
The Fab Four
Four (semi) symmetric zero series proved of special interest. They are the primus U(0,1)F, the secundus U(2,F)F, the tertius U(1,1)F and the quartus U(-1,1)F.
The set of integers is generated by U(0,1)2 or the primus of 2.
The set of odd integers is generated by U(-1,1)2 or the quartus of 2.
The Fibonacci series (U0=1, U1=2, Un+1 = Un + Un-1) consists of the alternating terms of the tertius and the primus of 3.
The Lucas series (U0=1, U1=3, Un+1 = Un + Un-1) consists of the alternating terms of the quartus and the secundus of 3.
The structure of the series of successively better approximations of the roots of natural numbers, revealed
Unsuspected patterns popped up, resulting in the discovery of the same series governing the successively closer rational approximations of the roots of natural numbers. A precise division of the set of non-square natural numbers in two distict classes emerged, for which Ed van Zon coined the names carréphylic and carréphobic.
Index correlated factorization
Other interesting phenomena are 'index correlated factorization' and the 'near-multiple' form that factors on prime indices show when 'sum-indexing' is applied to the tertius and quartus: Factors of terms show a one-to-one collelation with the factors of their index, and on prime indices the prime factors of the term, whether the term be prime or composite, are always 'near-multiples' of the form 2k*index±1. This is as yet unexplained.
Four (semi) symmetric zero series proved of special interest. They are the primus U(0,1)F, the secundus U(2,F)F, the tertius U(1,1)F and the quartus U(-1,1)F.
The set of integers is generated by U(0,1)2 or the primus of 2.
The set of odd integers is generated by U(-1,1)2 or the quartus of 2.
The Fibonacci series (U0=1, U1=2, Un+1 = Un + Un-1) consists of the alternating terms of the tertius and the primus of 3.
The Lucas series (U0=1, U1=3, Un+1 = Un + Un-1) consists of the alternating terms of the quartus and the secundus of 3.
The structure of the series of successively better approximations of the roots of natural numbers, revealed
Unsuspected patterns popped up, resulting in the discovery of the same series governing the successively closer rational approximations of the roots of natural numbers. A precise division of the set of non-square natural numbers in two distict classes emerged, for which Ed van Zon coined the names carréphylic and carréphobic.
Index correlated factorization
Other interesting phenomena are 'index correlated factorization' and the 'near-multiple' form that factors on prime indices show when 'sum-indexing' is applied to the tertius and quartus: Factors of terms show a one-to-one collelation with the factors of their index, and on prime indices the prime factors of the term, whether the term be prime or composite, are always 'near-multiples' of the form 2k*index±1. This is as yet unexplained.
And more
For what it's worth: using the operation 'sigma-repeated' - formalized by Ed van Zon - we've constructed the formula to find the entire polynome for any term of U(a,b,c)F.
The method used for most of this research was 'digging for data & prying for patterns'. Some of the patterns, though obvious enough, completely elude the revealers' understanding. They include 'index-correlated factorization', the 'near multiple' phenomenon and the phenomenon of 'series splitting' which emerged at some point, and involves a totally weird operation, guaranteed to have no application whatsoever.
Ed van Zon provided some excellent tools for the digging. Without them few of the results would ever have been found. He also doubled as an invaluable sounding board.
christian freeling