My favorite integer sequence

[jaydfox]

My favorite integer sequence is http://oeis.org/A018819

or resp http://oeis.org/A000123

1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60 etc

the idea that they (A018819) are computed as f(1)=f(2)=1 , f(2m-1) = f(1)+f(2)+...f(m) = f(2m) fascinates me.

This is a nicer sequence/equation than Fibonacci and Somos or even Sylvester imho.

Wow, that's a very cool sequence!

Also very appealing is that the second difference of any of these is itself and the first difference is the other one.
This seems like a natural analogue or extension of D exp(x) = exp(x) or the hyperbolic case (sinh,cosh) or even the differences of 2^n are 2^n.

....

I wonder if this can be extended naturally to a real-analytic function !?
I assume this is already done if possible , but I cant recall immediately.

Perhaps the equation Dif^[2] (f(z)) = f(z+q) helps.
(Dif is the difference operator and q is a real number)

Generalizations of this are awesome too.

You're right, I think we can generalize to the reals:
f'(x) = f(x/2) for real x

Rewriting as f'(2*x) = f(x), and taking repeated derivatives, you get:

\( \frac{d^n}{{dx}^n} f(2^{n} x) = f(x) \)

This becomes a differential equation. We only need the first derivative to solve numerically (small step size is needed), but a neat recurrence relation leads to the following power series:

\( f(x) = \sum_{k=0}^{\infty}\frac{1}{2^{k(k-1)/2} k!} x^k \)

Interestingly, the coefficients shrink a lot faster than the coefficients of exp(x), so this function should be entire. I haven't played with complex inputs yet, but I assume they will yield some additional strange insights. Anyway...

Solving the first order DE numerically with step size 2^-16, and comparing against the power series, I get this:

Code:

k   Seq(k)  f(k), solved numerically                 f(k) from taylor series
0    1      1.0000000000000000000000000000000000000  1.0000000000000000000000000000000000000
1    2      2.2714817776328560750431103337873690854  2.2714925555010614874285405888176164401
2    4      4.1773173941606529245612490821348973409  4.1773464748074343373543678899643338246
3    6      6.8671860897249745801245386620145375421  6.8672430206305994112385590729778566375
4    10     10.508411896004350162982061305231037214  10.508508500609246317119590891681491159
5    14     15.287018078186288230449529905529141263  15.287168724886801285376150072786915911
6    20     21.408813542320259733759350273849014727  21.409035429550209943052567297038082745
7    26     29.100511698013492345429157251156602689  29.100825155899256358829011727309468536
8    36     38.610882270215116543753385399927680549  38.611311079299434870920914930947641656
9    46     50.211936557632561352592030250666009907  50.212508285169806454977540792592286954
10   60     64.200146639137163586448535446092545747  64.200892993470395299080619414195121784
11   74     80.897699033343757431951513376485935868  80.898656236881542416713121934567339827
12   94     100.65378332039157982318289463496211517  100.65499250171026016241191203021809929
13   114    123.84591623881118998358511675859249968  123.84742384441605306292982511459497756
14   140    150.88130177423428738305958529789698736  150.88316000052091511117724764967070353
15   166    182.19822776059034989484940054653606685  182.20049500655531029093952174339991253
16   202    218.26749951833593186029040720103238278  218.27024085959392033522157069063040289
17   238    259.59391105817928883583638214653162175  259.59719874285183032581376577355118567
18   284    306.71775438269476078916183427557721551  306.72166834974364749998721226998604896
19   330    360.21636742216807724080306836388059879  360.22099584275484035373513994731204098
20   390    420.70572114497547515385049167763838108  420.71116098743637265925519388330259454

Pretty dang cool if you ask me. If we take k very high, the continuous function seems to converge on a constant multiple of the discrete sequence. I'm not sure what to make of this constant yet, but I assume it has an interesting interpretation. Based on analysis of the first 2^18 terms in the sequence, the constant is somewhere between 1.083035 and 1.083089, and probably pretty close to the center of that interval, about 1.083062.

Here is SAGE (python) code:

Code:

RF = RealField(128)
PR.<z> = PowerSeriesRing(RF)
fp = [1/(2^(k*(k-1)/2) * factorial(k)) for k in xrange(50)]
func = PR(fp)
for k in xrange(21):
    print "{0:2d} ".format(int(k)), func(k)

And here is the output of that code:

Code:

0  1.0000000000000000000000000000000000000
1  2.2714925555010614874285405888176164401
2  4.1773464748074343373543678899643338246
3  6.8672430206305994112385590729778566375
4  10.508508500609246317119590891681491159
5  15.287168724886801285376150072786915911
6  21.409035429550209943052567297038082745
7  29.100825155899256358829011727309468536
8  38.611311079299434870920914930947641656
9  50.212508285169806454977540792592286954
10  64.200892993470395299080619414195121784
11  80.898656236881542416713121934567339827
12  100.65499250171026016241191203021809929
13  123.84742384441605306292982511459497756
14  150.88316000052091511117724764967070353
15  182.20049500655531029093952174339991253
16  218.27024085959392033522157069063040289
17  259.59719874285183032581376577355118567
18  306.72166834974364749998721226998604896
19  360.22099584275484035373513994731204098
20  420.71116098743637265925519388330259454