Difficult Sequences

[rocdocmac]

Replace the ? in each of these sequences with the correct value ...

[1]  1  4  17  54  145  368  945  ?

[2]  0  6  21  405  -504  ?

[3]  2  15  10001  215441  ?

[4]  7  8  5  3  9  8  1  6  3  ? 

[5] 15  52  99  144  175  180  147  ?

[6]  0  5  8  8  2  3  5  2  9  4  ?

[7]  1   2r    πr²   ⁴/₃ πr³  ?   

[8]  4/10  ?/10² 168/10³  1229/10⁴  9592/10⁵  78498/10⁶

[9]  3  23  229  2869  43531  ?

[10]  14  21  13  2  5  8  0  19  5  18  9  5  ?

[11]  6  8  5  8  4  0  7  3  4  6  ?

[12]  1  3  8  22  65  209  732  2780  ?

Edited December 17, 2017 by rocdocmac
No title!

[bonanova]

21 hours ago, rocdocmac said:

Replace the ? in each of these sequences with the correct value ...

1.   1  4  17  54  145  368  945  ?
2.   0  6  21  405  -504  ?
3.   2  15  10001  215441  ?
4.   7  8  5  3  9  8  1  6  3  ? 
5.   15  52  99  144  175  180  147  ?
6.   0  5  8  8  2  3  5  2  9  4  ?
7.   1   2r    πr²   ⁴/₃ πr³ 1/₂ π²r⁴    { Can be written: r⁰  2r¹  πr^{2  4}/₃ πr^{3  1}/₂ π²r⁴ ... }
8.   4/10  ?/10² 168/10³  1229/10⁴  9592/10⁵  78498/10⁶
9.   3  23  229  2869  43531  ?
10.   14  21  13  2  5  8  0  19  5  18  9  5  ?
11.   6  8  5  8  4  0  7  3  4  6  ?
12.   1  3  8  22  65  209  732  2780  ?

 

[rocdocmac]

Spoiler

[7] Yes, a relatively easy one!

The volume of a glome (4D sphere).

Next one would be ⁸/₁₅ π²r ⁵  

 

[bonanova]

12 hours ago, rocdocmac said:

  Reveal hidden contents

[7] Yes, a relatively easy one!

The volume of a glome (4D sphere).

Next one would be ⁸/₁₅ π²r ⁵  

 

Spoiler

More precisely, volumes enclosed by nD spheres, or of nD balls.

Still, I'm wondering about the first term. Does it imply the volume enclosed by a point is 1? Should the first term be 0 r⁰ ?

 

[rocdocmac]

Spoiler

I've been wondering this too! Negative dimension?

A one-dimensional ball of radius r is a line segment of length 2r and its zero-dimensional sphere consists of two points separated by a distance of 2r. Therefore, the series probably only starts at 1, i.e. the "volume" of a zero-dimentional ball. 

 

[rocdocmac]

Spoiler

Let's rather compare this way ...

0D: vol = 1, area = 0

1D: vol = 2r, area = 2

2D: vol = πr² , area = 2πr

3D: vol =  ⁴/₃ πr³ , area =  4πr²

^etc.

^{Thus, for volume:}

1   2r   πr²   ⁴/₃ πr³  ...

but for bounding area:

0   2   2πr   4πr²   ...