[seqfan] Re: Pondering on Yet another infinite set (family) of sequences

[Peter Munn]

Hello Peter,

I think your (3,2) sequence is alternate terms of A001687 a(n) = a(n-2) +
a(n-5), which starts 1,0,1,0,1,1,1,2,1,3,2,4,4,5,7.  Your terms are sums
of diagonals incident on the first column of the triangle as displayed,
the others sums of diagonals incident on the second.

It seems worth considering whether any other diagonal sums may likewise be
present in sequences as alternate terms (or in other such ways).

Regards,

Peter M

On Sat, 27 May, 2017 7:14 pm, Peter Lawrence wrote:
> 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)
>
>