[seqfan] Re: Sequence related to A180919

[Charles Greathouse]

On general principle: Be sure when creating these (polynomial) entries
to include a link to the
Index entries for sequences related to linear recurrences with
constant coefficients
and to include keyword:easy.

> How would you complete and continue this sequence?

A sequence for 'number of squares by h-value' and a sequence for
'least h with exactly n squares or 0 of none exists' would be nice.
I'm not sure what the best method is for populating these sequences,
though; sorry.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Jan 17, 2011 at 6:00 PM, Bruno Berselli <berselli.bruno at yahoo.it> wrote:
> Dear Sequence Fans,
>
>                                          with reference to comment in A180919, I’m considering families made up by sequences of integers of the type “n^2+h*n+1”.
>
> If h=1, the sequence of numbers obtained by placing successively n=0,1,2,3,4,5,6,... contains only one square.
> Also for h=3,4,6,... the corresponding sequences contain only 1 square. Therefore, I have a family of sequences which contains only one square and 1 is the smallest “h” to be found in that set: though at the moment I have no idea of what all the other “h” could be, 1 is for sure the smallest.
>
> Likewise, I can identify a family of sequences of integers of the type “n^2+h*n+1” which contain however exactly 2 squares. For h=5,8,9,10,12,... there are sequences with this feature and in the family containing them all, 5 is of course the smallest “h” to be found.
>
> Let’s move on now to the sequences of integers of the type “n^2+h*n+1” which contain exactly 3 squares. For h=7,11,14,16,18,... there are sequences with this feature and in the family gathering them, 7 is the smallest “h” to be found.
>
> Once again, let’s consider the sequences of integers of the type “n^2+h*n+1” which contain exactly 4 squares. For h=13,17,19,22,25,... there are sequences with this feature and therefore, within the family they belong to, 13 is the smallest “h” to be found.
>
> At this rate, I’ve reached the following sequence made up by smaller “h” values – within brackets you see the number of squares identifying that family of sequences:
>
>
> (1) 1
> (2) 5
> (3) 7
> (4) 13
> (5) 66
> (6) 23
> (7) 731
> (8) 37
> (9) 47
> (10) 62
> (11)           ???
> (12) 82
> (13)           ???
> (14) 258
> (15) 194
> (16) 158
> (17)           ???
> (18) 142
> (19)           ???
> (20) 446
> (21) 254
> (22)           ???
> (23)           ???
> (24) 317
> (25) 322
> .
> .
> .
>
>
> I hope everything is correct. I must admit that this sequence strikes me a lot. It contains numbers which are rather big compared to those that can be found in the proximity and this leads me to think that even bigger numbers may come out.
>
> How would you complete and continue this sequence?
>
>
> Best greetings to all.
>
> Thanks ;)
>
>
>
> Bruno
>
>
>
>
>
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