# [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)

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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.

```