[seqfan] Pondering on Yet another infinite set (family) of sequences
Peter Lawrence
peterl95124 at sbcglobal.net
Sat May 27 20:14:56 CEST 2017
Neil,
Using (a,b) as the slope for a diagonal row sum of
Pascal's Triangle has been used before as in
V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers <http://www.fq.math.ca/Scanned/2-4/harris.pdf>, Fib. Quart. 2 (1964) 277-289, sequence u(n,3,1).
And this gets referenced sometimes, but not always, in the corresponding OEIS sequences,
So the first, simplest, question is should we uniformly add the reference and (a,b) info to a sequence belonging to this family ?
But then I get some sequences that aren’t already in the OEIS,
I think that previously only a <= b were considered,
should we add the new ones in ?
And how far down the Farey seq of (a,b), and their reciprocals (b,a), should we go ?
using (1,1) we get the unmodified Fibonacci seq,
for example the Pascal’s Triangle numbers on the 10’th 1:1 diagonal,
here marked with “+”, sum to the 10’th Fibonacci term
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1+
1 6 15 20 15+ 6 1
1 7 21 35+ 35 21 7 1
1 8 28+ 56 70 56 28 8 1
1 9+ 36 84 126 126 84 36 9 1
1+ 10 45 120 210 252 210 120 45 10 1
(0, 1): 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144 A000079
(1, 3): 1, 1, 1, 1, 2, 5, 11, 21, 37, 64, 113, 205, 377, 693, 1266, 2301, 4175, 7581, 13785, 25088, 45665 A003522
(1, 2): 1, 1, 1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572 A005251
(2, 3): 1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 58, 92, 149, 250, 431, 750, 1299, 2227, 3784, 6401, 10828 A137356
(1, 1): 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 A000045
(3, 2): 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 23, 34, 52, 81, 126, 194, 296, 450, 685, 1046, 1601 ???
(2, 1): 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278 A000930
(3, 1): 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 95, 131, 181, 250, 345 A003269
(1, 0): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 A000027
(Note that some of these are already in OEIS not because they are in the (a,b)-Fibonacci family,
But rather because they have simple linear recurrence formulas, yet another infinite set of sequences,
That overlaps with this one because all (a,b)-Fibonacci sequences have (simple?) linear recurrence formulas)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Neil,
As a computer scientist (applied math IMHO) by day, I cannot help but ponder about this situation at night:
There are an infinite number of {infinite set of sequences, the family being generated by parameters a,b,c,...}
So the logical question is are there effective algorithms for recognizing members of any of these families of sequences ?
And do we want to implement any of them in the OEIS search ? (Either as browser-based java apps, or as backend server apps ?)
So that we don’t have to enter in an infinite number of sequences into OEIS in order for someone to look one up !
Sincerely,
Peter Lawrence.
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