[seqfan] Sequence related to A180919

[Bruno Berselli]

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