[seqfan] Re: 3D version of A000938: 3-in-line inside the nXnXn cube

Sun May 23 14:50:07 CEST 2010

Going up through 4 extra planes for 3-6 collinear points, the current snapshot is

3 points in a side-3 x-dimensional grid

stat203.txt 8
stat303.txt 49
stat403.txt 272
stat503.txt 1441
stat603.txt 7448
stat703.txt 37969
stat803.txt 192032
stat903.txt 966721

4 points in a side-4 x-dimensional grid

stat204.txt 10
stat304.txt 76
stat404.txt 520
stat504.txt 3376
stat604.txt 21280
stat704.txt 131776
stat804.txt 807040
stat904.txt 4907776

5 points in a side-5 x-dimensional grid

stat205.txt 12
stat305.txt 109
stat405.txt 888
stat505.txt 6841
stat605.txt 51012
stat705.txt 372709
stat805.txt 2687088
stat905.txt 19200241

6 points in a side-6 x-dimensional grid

stat206.txt 14
stat306.txt 148
stat406.txt 1400
stat506.txt 12496
stat606.txt 107744
stat706.txt 908608
stat806.txt 7548800
stat906.txt 

3 points in a side-4 x-dimensional grid

stat203.txt 44
stat303.txt 376
stat403.txt 2960
stat503.txt 22624
stat603.txt 171584
stat703.txt 1303936
stat803.txt 9969920
stat903.txt 76793344

4 points in a side-5 x-dimensional grid

stat204.txt 64
stat304.txt 629
stat404.txt 5632
stat504.txt 48485
stat604.txt 410944
stat704.txt 3470549
stat804.txt 29389312
stat904.txt 250334405

5 points in a side-6 x-dimensional grid

stat205.txt 88
stat305.txt 984
stat405.txt 9952
stat505.txt 96096
stat605.txt 907648
stat705.txt 8494464
stat805.txt 79355392
stat905.txt 743241216

6 points in a side-7 x-dimensional grid

stat206.txt 116
stat306.txt 1459
stat406.txt 16520
stat506.txt 177727
stat606.txt 1861436
stat706.txt 19230379
stat806.txt 197501840
stat906.txt 

3 points in a side-5 x-dimensional grid

stat203.txt 152
stat303.txt 1858
stat403.txt 21680
stat503.txt 253690
stat603.txt 3023432
stat703.txt 36785458
stat803.txt 455700320
stat903.txt 5725140970

4 points in a side-6 x-dimensional grid

stat204.txt 234
stat304.txt 2820
stat404.txt 31176
stat504.txt 333840
stat604.txt 3546144
stat704.txt 37807680
stat804.txt 406924416
stat904.txt 4432154880

5 points in a side-7 x-dimensional grid

stat205.txt 364
stat305.txt 4833
stat405.txt 58360
stat505.txt 676941
stat605.txt 7731364
stat705.txt 88086873
stat805.txt 1008580720
stat905.txt 11652595461

6 points in a side-8 x-dimensional grid

stat206.txt 536
stat306.txt 7816
stat406.txt 103040
stat506.txt 1296928
stat606.txt 15977216
stat706.txt 
stat806.txt 
stat906.txt 

3 points in a side-6 x-dimensional grid

stat203.txt 372
stat303.txt 5696
stat403.txt 85584
stat503.txt 1324160
stat603.txt 21306432
stat703.txt 354876416
stat803.txt 6065958144
stat903.txt 105562511360

4 points in a side-7 x-dimensional grid

stat204.txt 660
stat304.txt 9767
stat404.txt 135240
stat504.txt 1856075
stat604.txt 25897500
stat704.txt 371453327
stat804.txt 5493496080
stat904.txt 83634563315

5 points in a side-8 x-dimensional grid

stat205.txt 1120
stat305.txt 17216
stat405.txt 241600
stat505.txt 3270656
stat605.txt 43790080
stat705.txt 587503616
stat805.txt 7955368960
stat905.txt 

6 points in a side-9 x-dimensional grid

stat206.txt 1824
stat306.txt 30252
stat406.txt 455040
stat506.txt 6558084
stat606.txt 92868384
stat706.txt 
stat806.txt 
stat906.txt 

3 points in a side-7 x-dimensional grid

stat203.txt 824
stat303.txt 16427
stat403.txt 330032
stat503.txt 6962675
stat603.txt 154462184
stat703.txt 3564072827
stat803.txt 84525491552
stat903.txt 2041422379715

4 points in a side-8 x-dimensional grid

stat204.txt 1524
stat304.txt 26272
stat404.txt 429360
stat504.txt 7075840
stat604.txt 120924864
stat704.txt 2166664192
stat804.txt 40707459840
stat904.txt 797134274560

5 points in a side-9 x-dimensional grid

stat205.txt 2876
stat305.txt 50778
stat405.txt 825560
stat505.txt 13092246
stat605.txt 208161236
stat705.txt 3369273138
stat805.txt 55993694000
stat905.txt 

6 points in a side-10 x-dimensional grid

stat206.txt 5100
stat306.txt 94800
stat406.txt 1602000
stat506.txt 26009280
stat606.txt 416136000
stat706.txt 
stat806.txt 
stat906.txt 


 rhhardin at mindspring.com
rhhardin at att.net (either)


----- Original Message ----
> From: Richard Mathar <mathar at strw.leidenuniv.nl>
> To: seqfan at seqfan.eu
> Sent: Sun, May 23, 2010 8:13:27 AM
> Subject: [seqfan] Re: 3D version of A000938: 3-in-line inside the nXnXn cube
> 
> 
Counted in

> href="http://list.seqfan.eu/pipermail/seqfan/2010-May/004752.html" target=_blank 
> >http://list.seqfan.eu/pipermail/seqfan/2010-May/004752.html :

rh> 
> Ron Hardin
rh> Sun May 23 12:55:25 CEST 2010
rh> 3 points in a 
> side-4 2..9 dimensional grid
rh> $ awk '{print FILENAME,$4}' 
> stat??3.txt
rh> stat203.txt 44
rh> stat303.txt 376
rh> 
> stat403.txt 2960
rh> stat503.txt 22624
rh> stat603.txt 
> 171584
rh> stat703.txt 1303936
rh> stat803.txt 9969920
rh> 
> stat903.txt 76793344

This looks like
a(n) = 
> 6^n+8^n/2-4^n*3/2,
a(n)= +18*a(n-1) -104*a(n-2) +192*a(n-3), 
> 
Extrapolation (offset 0 by adding a(0)=0 and 
> a(1)=4):
0,4,44,376,2960,22624,171584,1303936,9969920,76793344,595764224,4651472896,
36511354880,287837937664,2276984766464,18060760416256
g.f. 
> 4*x*(-1+7*x)/((6*x-1)*(8*x-1)*(4*x-1)).

rh> 4 points in a side-5 2..9 
> dimensional grid
rh> 
rh> $ awk '{print FILENAME,$5}' 
> stat??4.txt
rh> stat204.txt 64
rh> stat304.txt 629
rh> 
> stat404.txt 5632
rh> stat504.txt 48485
rh> stat604.txt 
> 410944
rh> stat704.txt 3470549
rh> stat804.txt 29389312
rh> 
> stat904.txt 250334405

This looks like
a(n) = 
> 3*7^n/2-2*5^n+9^n/2.
a(n)= +21*a(n-1) -143*a(n-2) 
> +315*a(n-3),
Extrapolation (offset 0 by adding a(0)=0, a(1)=5 which makes 
> sense):
0,5,64,629,5632,48485,410944,3470549,29389312,250334405,2147573824,18558863669,
161488417792,1413825023525,12443523805504,110005873155989
G.f. 
> of sequence if a(0) and a(1) included: 
> x*(-5+41*x)/((9*x-1)*(7*x-1)*(5*x-1)).


>From a grand point of view, 
> adding dimensions to these counting problems is
always susceptible to 
> combinatorial arguments of the form "If we add a layer
or a sub-plane, how 
> many new combinations are created by allowing some points
in that extra layer 
> or plane?" So linear recurrences are not a big 
> surprise.

RJM


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