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

[rkg]

See also The Leaning Tower of Pingala.

On Wed, 31 May 2017, Peter Munn wrote:

> 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??+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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