Magic Series | Formulae |
Number N(n, m) of magic series for hypercubes of dimension n and order m |
The formulae for small orders m and random dimensions n |
Order m = 2 | |||
Order m = 3 | |||
Order m = 4 | |||
Order m = 5 | |||
Order m = 6 |
Orders 7 to 11 |
The first coefficients of the formulae |
The sequence of the numerators |
1; 2; 8; 46; 352; 3364; 38656; 519446; 7996928; 138826588; ? |
...; 2683604992; 57176039628; 1331300646912; 33636118326984; 916559498182656; ... |
"All this is the proof that your coefficients are 100% correct and that they have a strong mathematical basis. But... it will be perhaps difficult to explain WHY this sequence and formula is used for hypercubes." |
Cloître's sequence: |
The surprising proof |
"I think I have now actually found a proof, for the case that m is even, that N(x,m) is indeed a quasi-polynomial
of degree m-1 in x, and that its coefficient of degree m-1 is indeed equal to f(m)/(2^(m-1) (m-1)!^2 ),
where f(m) is the m-th term in the sequence defined in OEIS A099765. My proof is based on geometric concepts, and in my mind it gives a clear geometric interpretation of all the terms and factors appearing in the expanded formula (with f(m) = the summation formula given in OEIS A099765). I also prove that, for the case that m is even and m≥4, the period L(m) of the quasi-polynomial N(x,m) in x is a divisor of l(m-1), where l(n) is the least common multiple of the first n natural numbers (1,2,…,n). All this seems to be true if m is odd as well, but adaptations of the current proof would certainly be necessary, if at all possible." |
What about the other coefficients? |
Approximation of N(n,m) for high orders m and dimensions n |
Thus we get good approximations of N(n, m) by just calculating the first summand of the affiliated formula. |
Let's speak about 'm1-approximation' (First Member Approximation). |
⇒ N(100,100) = 1.47857923327431013915144303218931705714497824542031... · 10^{19641} |
Comparisons in dimension n=2 |
Method | m = 10 | m = 100 | m = 1000 | m = 10000 |
Gerbicz, Kinnaes (exact) | 78132541528 | 9.0430074 · 10^{236} | 6.5918292254 · 10^{3424} | - |
Bottomley (approx.) | 78120036444 | 9.0430064 · 10^{236} | 6.5918292275 · 10^{3424} | 2.945158 · 10^{44330} |
m1-approximation | 78371381680 | 9.0466982 · 10^{236} | 6.592103 · 10^{3424} | 2.945170 · 10^{44330} |
The table confirms the different methods impressively. |
Magic series for cubes (n=3) |
For example: N_{3}(100) ≈ 1.4713509428 · 10^{435} (Expected relative error < 10^{-10} ) |
Overview |
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