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

Richard Mathar mathar at strw.leidenuniv.nl
Sun May 23 17:53:23 CEST 2010

```http://list.seqfan.eu/pipermail/seqfan/2010-May/004755.html

rh> 3 points in a side-3 x-dimensional grid
rh>
rh> stat203.txt 8
rh> stat303.txt 49
rh> stat403.txt 272
rh> stat503.txt 1441
rh> stat603.txt 7448
rh> stat703.txt 37969
rh> stat803.txt 192032
rh> stat903.txt 966721

This could be integrated in A005059:

%I A005059
%C A005059 Conjecture (verified up to a(9)): Number of collinear point triples in a 3 X 3 X 3 X... n-dimensional cubic grid [Ron Hardin (rhhardin(AT)att.net), May 23 2010]
%F A005059 a(2) = A000938(3). a(3) = A157882(3).

rh> 4 points in a side-4 x-dimensional grid
rh>
rh> stat204.txt 10
rh> stat304.txt 76
rh> stat404.txt 520
rh> stat504.txt 3376
rh> stat604.txt 21280
rh> stat704.txt 131776
rh> stat804.txt 807040
rh> stat904.txt 4907776

This would go into A081199:

%I A081199
%C A081199 Conjecture (verified up to a(9)): Number of collinear 4-tuples of points in a 4 X 4 X 4 X... n-dimensional cubic grid [Ron Hardin (rhhardin(AT)att.net), May 23 2010]

rh> 5 points in a side-5 x-dimensional grid
rh>
rh> stat205.txt 12
rh> stat305.txt 109
rh> stat405.txt 888
rh> stat505.txt 6841
rh> stat605.txt 51012
rh> stat705.txt 372709
rh> stat805.txt 2687088
rh> stat905.txt 19200241

This is good for A081200:

%I A081200
%C A081200 Conjecture (verified up to a(9)): Number of collinear 5-tuples of points in a 5 X 5 X 5 X... n-dimensional cubic grid [Ron Hardin (rhhardin(AT)att.net), May 23 2010]

rh> 6 points in a side-6 x-dimensional grid
rh>
rh> stat206.txt 14
rh> stat306.txt 148
rh> stat406.txt 1400
rh> stat506.txt 12496
rh> stat606.txt 107744
rh> stat706.txt 908608
rh> stat806.txt 7548800
rh> stat906.txt

Could go into A081201:

%I A081201
%C A081201 Conjecture (verified up to a(8)): Number of collinear 6-tuples of points in a 6 X 6 X 6 X... n-dimensional cubic grid [Ron Hardin (rhhardin(AT)att.net), May 23 2010]

rh> 3 points in a side-4 x-dimensional grid
rh>
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 is new, I guess (Template. I will NOT submit these...)

%I A000001
%S A000001 0,4,44,376,2960,22624,171584,1303936,9969920,76793344
%N A000001 Number of collinear point triples in a 4 X 4 X 4 X... n-dimensional cubic grid
%F A000001 Conjecture: a(n) = 8^n/2-3*4^n/2+6^n. a(n)= +18*a(n-1) -104*a(n-2) +192*a(n-3). G.f.: 4*x*(-1+7*x)/((6*x-1)*(8*x-1)*(4*x-1)). [R. J. Mathar, May 23 2010]
%Y A000001 Cf. A005059.
%K A000001 nonn
%O A000001 0,2
%A A000001 Ron Hardin (rhhardin(AT)att.net), May 23 2010

rh> 4 points in a side-5 x-dimensional grid
rh>
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

Another candidate:

%I A000002
%S A000002 0,5,64,629,5632,48485,410944,3470549,29389312,250334405
%N A000002 Number of collinear point 4-tuples in a 5 X 5 X 5 X... n-dimensional cubic grid
%F A000002 Conjecture: a(n) = 9^n/2+3*7^n/2-2*5^n. a(n)= +21*a(n-1) -143*a(n-2) +315*a(n-3). G.f.: x*(-5+41*x)/((9*x-1)*(7*x-1)*(5*x-1)). [R. J. Mathar, May 23 2010]
%K A000002 nonn
%O A000002 0,2
%A A000001 Ron Hardin (rhhardin(AT)att.net), May 23 2010

rh> 5 points in a side-6 x-dimensional grid
rh>
rh> stat205.txt 88
rh> stat305.txt 984
rh> stat405.txt 9952
rh> stat505.txt 96096
rh> stat605.txt 907648
rh> stat705.txt 8494464
rh> stat805.txt 79355392
rh> stat905.txt 743241216

%I A000003
%S A000003 0,6,88,984,9952,96096,907648,8494464,79355392,743241216
%N A000003 Number of collinear point 5-tuples in a 6 X 6 X 6 X... n-dimensional cubic grid
%F A000003 Conjecture: a(n) = 10^n/2+2*8^n-5*6^n/2. a(n)= 24*a(n-1) -188*a(n-2) +480*a(n-3). G.f.: 2*x*(-3+28*x)/((6*x-1)*(8*x-1)*(10*x-1)). [R. J. Mathar, May 23 2010]
%K A000003 nonn
%O A000003 0,2
%A A000003 Ron Hardin (rhhardin(AT)att.net), May 23 2010

rh> 6 points in a side-7 x-dimensional grid
rh>
rh> stat206.txt 116
rh> stat306.txt 1459
rh> stat406.txt 16520
rh> stat506.txt 177727
rh> stat606.txt 1861436
rh> stat706.txt 19230379
rh> stat806.txt 197501840
rh> stat906.txt

%I A000004
%S A000004 0,7,116,1459,16520,177727,1861436,19230379,197501840
%N A000004 Number of collinear point 6-tuples in a 7 X 7 X 7 X... n-dimensional cubic grid
%F A000004 Conjecture: a(n) = 11^n/2+5*9^n/2-3*7^n. a(n)= 27*a(n-1) -239*a(n-2) +693*a(n-3). G.f.: x*(-7+73*x)/((11*x-1)*(7*x-1)*(9*x-1)). [R. J. Mathar, May 23 2010]
%K A000004 nonn
%O A000004 0,2
%A A000004 Ron Hardin (rhhardin(AT)att.net), May 23 2010

rh> 3 points in a side-5 x-dimensional grid
rh>
rh> stat203.txt 152
rh> stat303.txt 1858
rh> stat403.txt 21680
rh> stat503.txt 253690
rh> stat603.txt 3023432
rh> stat703.txt 36785458
rh> stat803.txt 455700320
rh> stat903.txt 5725140970

It looks like the order of the recurrences increases by 1 if the difference between
the number of points and the edge length of the grid increase by 1:

%I A000005
%S A000005 0,10,152,1858,21680,253690,3023432,36785458,455700320,5725140970
%N A000005 Number of collinear point triples in a 5 X 5 X 5 X... n-dimensional cubic grid
%F A000005 Conjecture: a(n) = 7^n-5^(n+1)/2+13^n/2+9^n. a(n)= 34*a(n-1) -416*a(n-2) +2174*a(n-3) -4095*a(n-4). G.f.:  2*x*(5-94*x+425*x^2)/((9*x-1)*(13*x-1)*(7*x-1)*(5*x-1)). [R. J. Mathar, May 23 2010]
%K A000005 nonn
%O A000005 0,2
%A A000005 Ron Hardin (rhhardin(AT)att.net), May 23 2010

rh> 4 points in a side-6 x-dimensional grid
rh>
rh> stat204.txt 234
rh> stat304.txt 2820
rh> stat404.txt 31176
rh> stat504.txt 333840
rh> stat604.txt 3546144
rh> stat704.txt 37807680
rh> stat804.txt 406924416
rh> stat904.txt 4432154880

%I A000006
%S A000006 0,15,234,2820,31176,333840,3546144,37807680,406924416,4432154880
%N A000006 Number of collinear point 4-tuples in a 6 X 6 X 6 X... n-dimensional cubic grid
%F A000006 Conjecture: a(n) = 3*8^n+3*10^n/2-5*6^n+12^n/2. a(n)= 36*a(n-1) -476*a(n-2) +2736*a(n-3) -5760*a(n-4). G.f.:  3*x*(5-102*x+512*x^2)/((12*x-1)*(6*x-1)*(8*x-1)*(10*x-1)). [R. J. Mathar, May 23 2010]
%K A000006 nonn
%O A000006 0,2
%A A000006 Ron Hardin (rhhardin(AT)att.net), May 23 2010

```