[seqfan] Re: Pondering on Yet another infinite set (family) of sequences
Peter Munn
techsubs at pearceneptune.co.uk
Wed May 31 12:55:29 CEST 2017
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 arenâ€™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 Pascalâ€™s Triangle numbers on the 10â€™th 1:1 diagonal,
> here marked with â€œ+â€, sum to the 10â€™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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