# [seqfan] Re: Donald Knuth Turns Eighty

Hugo Pfoertner yae9911 at gmail.com
Wed Jan 17 23:18:31 CET 2018

```In the video of Don's Christmas Tree Lecture 2017 at
there is a comment by Konstantin Vladimirov, providing a link to a program
"naivepavings.cc" via GitHub
https://github.com/tilir/generators

I've used this program to compute a few more terms for A285357 and have put
the results into a new non-redundant triangular table at
https://oeis.org/draft/A298362
T(6,6) is the first not yet computed term, but I hope to get it with some
patience.

In Don's lecture he gives a hint
https://youtu.be/BxQw4CdxLr8?t=2653
on the leading terms of a formula for https://oeis.org/A285361 saying "I'm
gonna actually reveal more than I should ..."
a_{3,n} = (1/4)*(3^(n+3) - 5*2^(n+4) + ...?...)  ...?... = "this 3 terms".
I could not resist to try the simplest possible case, i.e., a quadratic
polynomial for "this 3 terms" and got a perfect match.

As Don mentioned that he got a confirmation that the problem has already
been solved successfully
https://youtu.be/BxQw4CdxLr8?t=2935
I suppose that it's ok to show it here:

A285361(n) = (1/4) * (3^(n+3) - 5*2^(n+4) + 4*n^2 + 26*n + 53)

The formula also predicts the same A285361(10)=378279 as computed by
Vladimirov's program. If this is confirmed, we could provide a b-file
1 1
2 11
3 64
4 282
5 1071
6 3729
7 12310
8 39296
9 122773
10 378279
11 1154988
12 3505542
13 10598107
14 31957661
15 96200098
16 289255020
17 869075073
18 2609845875
19 7834779640
20 23514823730
21 70565441671
22 211738266921
23 635298685614
24 1906063827672
25 5718527025901
26 17156252164799
27 51470098670020
28 154412980362846
29 463244309795763
30 1389743666803509
31 4169252475244858
32 12507800375405252
33 37523487025559257
34 112570632875367051
35 337712242223482128
36 1013137413865210890

Hugo Pfoertner

On Wed, Jan 10, 2018 at 5:34 PM, Peter Luschny <peter.luschny at gmail.com>
wrote:

> We could make him happy and calculate some more terms of
> https://oeis.org/A285357 and https://oeis.org/A285361
>
> Don suggests that one might look at a few conjectures about
> set partitions and generating functions that he has put online at
> http://www-cs-faculty.stanford.edu/~knuth/caspagf.txt
>
> And be sure to use triangular paper when you write your
> solutions down!  https://goo.gl/4twPnf
>
> Happy birthday, Don!
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>
```