[seqfan] Re: 4-analogue of Ulam numbers?

[Jonathan Post]

Thank you, Alois. It is worth pointing out at for the k-analogue of
Ulam numbers (the originals of which are 2-Ulam), we always appear to
have a square (k^2) after the triangular number k(k+1)/2:

k=2: 1, 2, 3, (4)
k=3: 1, 2, 3, 6,  (9)
k=4: 1, 2, 3, 4, 10, (16)
k=5: 1, 2, 3, 4, 5, 15, (25)
k=6: 1, 2, 3, 4, 5, 6, 21, (36)
k=7: 1, 2, 3, 4, 5, 6, 7, 28, (49)
k=8: 1, 2, 3, 4, 5, 6, 7, 8, 36, (64)
etc.

The reason is not clear to me. The sequence of numbers past the k^2 in
the k-analogue, A[k,n+3] in the array are:
6, 10, 17, 26, 37, 50, 65, 82, 101, seem to "want" to be k^2 + 1
unless nonuniquness is an obstruction.

How much of the analytical machinery of N. J. A. Sloane, David Wilson,
D. E. Knuth, Jud McCranie, J. Schmerl and E. Spiegel, M. C. Wunderlich
et al. apply to the other rows in the array?

Do the terms of A[k,n] lie very close to a straight line f(k,n) as
they do when k=2 and 13.51*n is the line?

After a few initial terms, do the sequences settle into a regular
pattern of dense clumps separated by sparse gaps, and if so, with what
periodicity?

I had the pleasure of speaking with Ulam, and the double sadness of
his dying before we could co-author the paper based on my PhD
dissertation results which interested him. Hence my thrill is probing
at one of his lesser-known but fascinating discoveries.

On Sun, Jan 2, 2011 at 7:09 AM, Alois Heinz <heinz at hs-heilbronn.de> wrote:
>
> Nice problem!
>
> The 4-analogue of Ulam numbers:
>
> 1, 2, 3, 4, 10, 16, 17, 18, 19, 22, 64, 65, 66, 68, 69, 128, 132, 188,
> 190, 191, 194, 252, 253, 255, 313, 314, 318, 374, 376, 377, 436, 441,
> 496, 497, 499, 500, 502, 560, 561, 563, 621, 622, 626, 682, 684, 685,
> 687, 745, 746, 805, 811, 865, 866, 867, 869, 870, 871, 930, 932, 990
>
> For every k>=2 there is a k-analogue of Ulam numbers:
>
> 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, ...
> 1, 2, 3, 6, 9, 10, 11, 12, 28, 29, 30, 53, 56, 57, 80, ...
> 1, 2, 3, 4, 10, 16, 17, 18, 19, 22, 64, 65, 66, 68, 69, ...
> 1, 2, 3, 4, 5, 15, 25, 26, 27, 28, 29, 35, 43, 45, 165, ...
> 1, 2, 3, 4, 5, 6, 21, 36, 37, 38, 39, 40, 41, 51, 61, ...
> 1, 2, 3, 4, 5, 6, 7, 28, 49, 50, 51, 52, 53, 54, 55, ...
> 1, 2, 3, 4, 5, 6, 7, 8, 36, 64, 65, 66, 67, 68, 69, ...
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 45, 81, 82, 83, 84, 85, ...
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 55, 100, 101, 102, 103, ...
>
> Best regards, Alois
>
> Am 01.01.2011 19:02, schrieb Jonathan Post:
>>
>> Much is known now about
>> A002858 Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least
>> number>  a(n-1) which is a unique sum of two distinct earlier terms.
>>
>> I've just had a comment approved of a 3-analogue of Ulam numbers:
>> A007086  Next term is uniquely the sum of 3 earlier terms.
>> COMMENTS
>> a(1)=1, a(2)=2, a(3)=3, for n>3, a(n) = least number which is a unique
>> sum of three distinct earlier terms.  Written this way, we see that
>> this is to 3 as Ulam number A002858 is to 2. - Jonathan Vos Post
>> (jvospost3(AT)gmail.com)
>>
>> The next sequence in the supersequence appears, based on quick work by
>> hand, to be:
>>
>> 4-analogue of Ulam numbers: a(1) = 1; a(2) = 2; a(3) = 3, a(4) = 4,
>> for n>4, a(n) = least number>  a(n-1) which is a unique sum of 4
>> distinct earlier terms.
>>
>> 1, 2, 3, 4, 10, 16, 17, 18, 19, 22, 63?
>>
>> n...a(n)
>> 1...1, by definition
>> 2...2, by definition
>> 3...3, by definition
>> 4...4, by definition
>> 5...10, because 1+2+3+4=10 uniquely, and none of {5,6,7,8,9} are in
>> the set of sums of 4 unique terms of {1,2,3,4} as that set is a
>> singlet.
>> 6...16, because 1+2+3+10 = 16, and none of {11,12,13,14,15} seem to be
>> in the set of sums of 4 unique terms of {1,2,3,4,10}
>> 7...17, based on back of the envelope, not confirmed
>> 8...18, ditto
>> 9...19, ditto
>> 10..22, I think
>> 11..63 maybe? 4+8+19+22 = 63; I've found witnesses to nonuniqueness of
>> 23 through 62
>>
>> This is not hard to code, as Ulam and Knuth did…
>>
>
>