[math-fun] Ulam(1,2)

[Klaus Brockhaus]

For 1000 <= n <= 1259, (12336 <= u(n) <= 15466), I get the following distribution of f(n), which seems to support your observation:

range of f(n)  count
[0, 0.050)      37
[0.050, 0.10)   25
[0.10, 0.15)    29
[0.15, 0.20)    22
[0.20, 0.25)    20
[0.25, 0.30)    11
[0.30, 0.35)    12
[0.35, 0.40)    7
[0.40, 0.45)    10
[0.45, 0.50)    1
[0.50, 0.55)    2
[0.55, 0.60)    0
[0.60, 0.65)    0
[0.65, 0.70)    0
[0.70, 0.75)    2
[0.75, 0.80)    6
[0.80, 0.85)    6
[0.85, 0.90)    24
[0.90, 0.95)    21
[0.95, 1.0)     25

Klaus

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> David Wilson wrote:
> 
> I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and including
> every subsequent number which is a unique sum of distinct earlier terms.  This
> is Sloane's A002858.
> 
> I took this sequence out quite a ways, and I noticed that after an initial flurry of
> numbers that are fairly uniform in distribution, the sequence starts to separate
> out into more or less regular clumps of numbers with a period that seems to
> be slighly more than 21.6.  Between these clumps are spaces containing
> relatively few numbers.
> 
> To see the phenomenon, let u(n) be the nth Ulam(1,2) number, and define
> 
>      f(n) = u(n) / 21.6 - [ u(n) / 2.16 ]
> 
> f(n) is a number on [0, 1) which indicates "u(n) mod 21.6".  If we compute
> f(n) for 1000 <= n <= 1999, (12336 <= u(n) <= 25511), we find the following
> distribution of f(n):
> 
>         range of f(n)  count
>         [0.00, 0.05)     110
>         [0.05, 0.10)     101
>         [0.10, 0.15)     120
>         [0.15, 0.20)      97
>         [0.20, 0.25)      90
>         [0.25, 0.30)      67
>         [0.30, 0.35)      58
>         [0.35, 0.40)      40
>         [0.40, 0.45)      31
>         [0.45, 0.50)       9
>         [0.50, 0.55)       4
>         [0.55, 0.60)       0
>         [0.60, 0.65)       0
>         [0.65, 0.70)       2
>         [0.70, 0.75)       7
>         [0.75, 0.80)      11
>         [0.80, 0.85)      22
>         [0.85, 0.90)      64
>         [0.90, 0.95)      66
>         [0.95, 1.00)     101
> 
> The values of f(n) are not uniformly distributed, the distribution
> indicate that u(n) values clump together with clumps occuring with a
> period of about 21.6.
> 
> This all presupposes that my computations of Ulma(1,2) are correct.
> Can anyone confirm my data?
> 
> Has this phenomenon been noted in the literature?
>