[seqfan] Re: Triangles of sums

Nacin, David NACIND at wpunj.edu
Fri Jul 23 19:51:17 CEST 2021


I used Python to grab the first hundred terms for the six variations:

Non-extendable up to reflection (This is Jonathan's initial sequence and it matches his data):

1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 9, 11, 11, 18, 17, 22, 23, 29, 31, 38, 37, 46, 49, 58, 59, 72, 76, 86, 90, 106, 115, 131, 140, 159, 177, 189, 204, 236, 254, 274, 292, 328, 355, 398, 404, 455, 485, 518, 555, 622, 647, 698, 727, 808, 837, 922, 939, 1032, 1100, 1161, 1198, 1320, 1363, 1480, 1511, 1644, 1706, 1844, 1884, 2042, 2109, 2254, 2319, 2534, 2599, 2778, 2848, 3068, 3180, 3389, 3463, 3726, 3859, 4114, 4219, 4530, 4660, 4942, 5081, 5453, 5612, 5960, 6090, 6535, 6708, 7121, 7321, 7804, 8013

Non-extendable all included:

1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 18, 22, 22, 36, 34, 44, 46, 58, 62, 76, 74, 92, 98, 116, 118, 144, 152, 172, 180, 212, 230, 262, 280, 318, 354, 378, 408, 472, 508, 548, 584, 656, 710, 796, 808, 910, 970, 1036, 1110, 1244, 1294, 1396, 1454, 1616, 1674, 1844, 1878, 2064, 2200, 2322, 2396, 2640, 2726, 2960, 3022, 3288, 3412, 3688, 3768, 4084, 4218, 4508, 4638, 5068, 5198, 5556, 5696, 6136, 6360, 6778, 6926, 7452, 7718, 8228, 8438, 9060, 9320, 9884, 10162, 10906, 11224, 11920, 12180, 13070, 13416, 14242, 14642, 15608, 16026

Max-height up to reflection:

1, 1, 1, 1, 2, 2, 3, 1, 3, 3, 5, 7, 9, 9, 16, 15, 20, 21, 27, 3, 1, 4, 5, 9, 9, 19, 19, 30, 30, 38, 47, 59, 66, 78, 88, 111, 114, 133, 157, 177, 188, 211, 1, 1, 1, 2, 4, 4, 5, 9, 11, 12, 17, 24, 30, 27, 42, 52, 59, 72, 93, 95, 124, 130, 154, 170, 211, 234, 263, 294, 335, 366, 417, 455, 508, 575, 658, 677, 778, 811, 934, 982, 1101, 1178, 1306, 1383, 1523, 1632, 1798, 1854, 2103, 2169, 2394, 2487, 2760, 2845, 3168, 1, 3595, 1

Max-height all included:

1, 1, 2, 2, 4, 4, 6, 2, 6, 6, 10, 14, 18, 18, 32, 30, 40, 42, 54, 6, 2, 8, 10, 18, 18, 38, 38, 60, 60, 76, 94, 118, 132, 156, 176, 222, 228, 266, 314, 354, 376, 422, 2, 2, 2, 4, 8, 8, 10, 18, 22, 24, 34, 48, 60, 54, 84, 104, 118, 144, 186, 190, 248, 260, 308, 340, 422, 468, 526, 588, 670, 732, 834, 910, 1016, 1150, 1316, 1354, 1556, 1622, 1868, 1964, 2202, 2356, 2612, 2766, 3046, 3264, 3596, 3708, 4206, 4338, 4788, 4974, 5520, 5690, 6336, 2, 7190, 2

Any-height up to reflection:

1, 1, 2, 2, 3, 3, 4, 5, 8, 8, 11, 13, 16, 16, 24, 23, 29, 30, 37, 42, 48, 50, 61, 67, 77, 86, 99, 110, 122, 132, 153, 166, 188, 199, 226, 250, 268, 287, 330, 351, 379, 403, 450, 479, 536, 545, 610, 647, 691, 736, 821, 850, 918, 962, 1061, 1090, 1207, 1238, 1353, 1443, 1534, 1574, 1743, 1796, 1948, 2005, 2184, 2277, 2452, 2520, 2728, 2833, 3029, 3123, 3414, 3520, 3761, 3866, 4171, 4305, 4630, 4725, 5090, 5274, 5635, 5755, 6201, 6373, 6796, 6968, 7477, 7662, 8188, 8334, 8976, 9165, 9774, 9974, 10684, 10922

Any-height all included:

1, 1, 3, 3, 5, 5, 7, 9, 15, 15, 21, 25, 31, 31, 47, 45, 57, 59, 73, 83, 95, 99, 121, 133, 153, 171, 197, 219, 243, 263, 305, 331, 375, 397, 451, 499, 535, 573, 659, 701, 757, 805, 899, 957, 1071, 1089, 1219, 1293, 1381, 1471, 1641, 1699, 1835, 1923, 2121, 2179, 2413, 2475, 2705, 2885, 3067, 3147, 3485, 3591, 3895, 4009, 4367, 4553, 4903, 5039, 5455, 5665, 6057, 6245, 6827, 7039, 7521, 7731, 8341, 8609, 9259, 9449, 10179, 10547, 11269, 11509, 12401, 12745, 13591, 13935, 14953, 15323, 16375, 16667, 17951, 18329, 19547, 19947, 21367, 21843

Note that the max-height variations are obviously not increasing sequences.  Would be interesting to know if they always return to 1 when they decrease or if there ends up being a point where there are multiple distinct triangles when a new height appears.  More interesting is the fact that there are no height-six pyramids for n=99 even though there are for n=98 and n=100.   The set of numbers for which the max height is less than it is for some previous number would be an interesting sequence.  Also, the set of numbers for which there is only one pyramid of max height up to reflection would be interesting. Both 103 and 104 have a unique height-six as well.  There are a lot of interesting angles to investigate.
 
-David




-----Original Message-----
From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of Allan Wechsler
Sent: Friday, July 23, 2021 11:43 AM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Triangles of sums

Now I understand. You only count "maximal" triangles, ones that cannot be extended by another row. (3) doesn't count because (3; 1 2) exists, and (9;
6 3) doesn't count because (9; 6 3; 5 1 2) exists.

Does the sequence counting *all* such triangles already exist? If it does, it should be linked.

I might write a program to see if I can confirm your numbers, now that I understand your rules.

On Fri, Jul 23, 2021 at 10:44 AM jnthn stdhr <jstdhr at gmail.com> wrote:

> Maybe this will help...
>
>   For each element in a triangle, consider all partitions of that 
> element that have exactly two parts (possible children).  For n=2, 
> place 2 at the apex to form a one-row triangle:
>
> 2
>
> Now try adding new row(s):
>
>   2
> 1 1
>
>   This is not valid and we have no more partitions to try, so a(2) = 1
>
>   But if all children can have distinct children then we have a new 
> and complete row and the triangle(s) of lesser height are "incomplete."
>
>   Let's look at possible triangles for a(9), which include the first
> *successful* use of backtracking:
>
>                                     9          9                     9
>         9      9      9        6 3       6 3       9        5 4
> 9    8 1   7 2   6 3    5 1 2    4 2 1    5 4     2 3 1
>
>   Because we *can* have more than one row the first triangle is dropped.
> The next two are valid and have no possible children, so we count them.
> The fourth one is dropped because the next two utilize backtracking to 
> find solutions with *more* than two rows.  The seventh one is dropped 
> because we find one solution with more than two rows.  So we end up with a(9) = 5:
>
>
>                     9          9          9
>   9      9      6 3       6 3       5 4
> 8 1   7 2   5 1 2    4 2 1    2 3 1
>
> Hope this helps.
>
> -jnthn
>
>
> On Fri, Jul 23, 2021, 6:26 AM jnthn stdhr <jstdhr at gmail.com> wrote:
>
> >
> >
> > On Fri, Jul 23, 2021, 6:13 AM Allan Wechsler <acwacw at gmail.com> wrote:
> >
> >> Okay, that list helps me focus on my area of confusion. Why do you
> exclude
> >> a singleton 3? You allow a singleton 2. What is the rule?
> >>
> >> On Fri, Jul 23, 2021, 5:40 AM jnthn stdhr <jstdhr at gmail.com> wrote:
> >>
> >> > Hi, Allan.
> >> >
> >> > To clarify, 1 and 2 have no possible distinct children, hence 
> >> > they
> have
> >> > height of 1.  And you are correct, I am not counting reflections,
> since
> >> > A340389 does not.
> >> >
> >> > The first few triangles are:
> >> >
> >> >             3      4      5      5
> >> > 1,  2,  1 2,  1 3,  1 4,  2 3
> >> >
> >> > As for the typo, In my notebook see I have the 10, 64, 513 
> >> > triangle
> just
> >> > below the 8, 53, 412 triangle, so I think my error is a result of
> >> looking
> >> > at the wrong line and not seeing the obvious error 6+4!=8. Sorry 
> >> > for
> the
> >> > confusion.
> >> >
> >> > -jnthn
> >> >
> >> >
> >> > On Thursday, July 22, 2021, Allan Wechsler <acwacw at gmail.com> wrote:
> >> >
> >> > > I applaud your instinct to make sure that simple cases 
> >> > > accompany
> their
> >> > more
> >> > > complicated brethren into the Encyclopedia -- I think this is 
> >> > > right
> on
> >> > > target.
> >> > >
> >> > > But I am missing something here. Can you display all the 
> >> > > triangles
> for
> >> > n=1
> >> > > to 3? My problem is that you must be allowing triangles of one 
> >> > > row
> in
> >> > order
> >> > > to have one example for n=1 and n=2, but then it seems to me 
> >> > > that
> you
> >> > ought
> >> > > to have two examples for n=3, one with one row, and one with 
> >> > > two
> rows.
> >> > But
> >> > > you say there is only one.
> >> > >
> >> > > Also, I am assuming you do *not* consider reflections around 
> >> > > the
> >> vertical
> >> > > axis to be distinct solutions.
> >> > >
> >> > > I'm sure some sequence fanatic will be happy to help you as 
> >> > > soon as
> >> it's
> >> > > clearer what your definitions are.
> >> > >
> >> > > One last thing: 6 + 4 does not equal 8, as your second example 
> >> > > seems
> >> to
> >> > > claim.
> >> > >
> >> > > On Thu, Jul 22, 2021 at 5:06 PM jnthn stdhr <jstdhr at gmail.com>
> wrote:
> >> > >
> >> > > > Hello seqfans.
> >> > > >
> >> > > > Long time no sequence (apologies.)  Inspired by ,
> >> > > https://nam11.safelinks.protection.outlook.com/?url=http%3A%2F%
> >> > > 2Foeis.org%2FA340389&data=04%7C01%7Cnacind%40wpunj.edu%7C92
> >> > > 60b4fb5d334dae090208d94dfab3b2%7C74540637643546cc87a46d38efb785
> >> > > 38%7C0%7C0%7C637626561419045200%7CUnknown%7CTWFpbGZsb3d8eyJWIjo
> >> > > iMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1
> >> > > 000&sdata=9ABXfr2O%2F7SCbPnkQve0%2F5Pcj2uvV6D4QFLKhTwzAEw%3
> >> > > D&reserved=0
> >> > > > wondered if a generalized sequence, the number of sum 
> >> > > > triangles of
> >> n,
> >> > > was
> >> > > > in the database -- it appears it is not.
> >> > > >
> >> > > > If we define a sum triangle of n as a triangle with n at its 
> >> > > > apex,
> >> all
> >> > > > pair-wise members (x, y) of rows 2,3,4,... sum to the element
> >> > immediately
> >> > > > above, every element is distinct, and rows are complete 
> >> > > > (length of
> >> row
> >> > m
> >> > > =
> >> > > > length of row (m-1) + 1.
> >> > > >
> >> > > > For example:
> >> > > >
> >> > > >           8         9        9
> >> > > >  3      6 4      6 3     6 3
> >> > > > 2 1   5 1 3   5 1 2  4 2 1
> >> > > >
> >> > > >
> >> > > > The sequence I get for n=1 to 30 is:
> >> > > >
> >> > > >
> >> > > > [1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 9, 11, 11, 18, 17, 22, 23, 
> >> > > > 29,
> 31,
> >> > 38,
> >> > > > 37, 46, 49, 58, 59, 72, 76, 86, 90]
> >> > > >
> >> > > > My python code is about 70 lines long.  Maybe a MMA expert 
> >> > > > could
> >> write
> >> > a
> >> > > > more concise program and confirm the the sequence?
> >> > > >
> >> > > > -Jonathan
> >> > > >
> >> > > > --
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> >> > > > fb78538%7C0%7C0%7C637626561419045200%7CUnknown%7CTWFpbGZsb3d8
> >> > > > eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6M
> >> > > > n0%3D%7C1000&sdata=H9JCM5hKZUsiGG0gzEz9yk2YFLKiactV83gmUn
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> >> > 7C0%7C637626561419045200%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAw
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