Fw: sequence A030444

[Joseph S. Myers]

On Thu, 20 May 2004, David Wilson wrote:

> Guenter Stertenbrink (Sterten at aol.com) gives me the following sequences for the number
> of fixed knight-move-connected polyominoes and fixed zebra-move-connected polyominoes
> (zebra = (2,3) leaper).  These polyominoes would not in general be connected in the usual
> edge-connected sense.
> 
> knight: 1,4,28,234,2162,20972,209608,2135572,22049959,229939414,2416816416 
> zebra: 1,4,28,234,2162,21272,218740,2323730,25314097,281299736
> 
> The knight sequence is a proposed correction to my sequence A030444.  At the present
> time I am unable to pursue verification.  Can someone verify Guenter's sequence to 4 or 5
> terms, and if correct, update A030444 and add the new sequence?

I don't know whether he computed these other sequences, but I put the
definitions of polyknights and polyzebras into my polyform enumeration
program generator and got the following figures that agree with the above.

Fixed polyknights (A030444):
polyknights[fixed][1] = 1
polyknights[fixed][2] = 4
polyknights[fixed][3] = 28
polyknights[fixed][4] = 234
polyknights[fixed][5] = 2162
polyknights[fixed][6] = 20972
polyknights[fixed][7] = 209608
polyknights[fixed][8] = 2135572
polyknights[fixed][9] = 22049959
polyknights[fixed][10] = 229939414
polyknights[fixed][11] = 2416816416

Free polyknights (A030446):
polyknights[free][1] = 1
polyknights[free][2] = 1
polyknights[free][3] = 6
polyknights[free][4] = 35
polyknights[free][5] = 290
polyknights[free][6] = 2680
polyknights[free][7] = 26379
polyknights[free][8] = 267598
polyknights[free][9] = 2758016
polyknights[free][10] = 28749456
polyknights[free][11] = 302120154

One-sided polyknights (A030445):
polyknights[one-sided][1] = 1
polyknights[one-sided][2] = 2
polyknights[one-sided][3] = 8
polyknights[one-sided][4] = 68
polyknights[one-sided][5] = 550
polyknights[one-sided][6] = 5328
polyknights[one-sided][7] = 52484
polyknights[one-sided][8] = 534793
polyknights[one-sided][9] = 5513338
polyknights[one-sided][10] = 57494308
polyknights[one-sided][11] = 604212943

Polyknights with bilateral symmetry (A030447):
polyknights[bilateral][1] = 1
polyknights[bilateral][2] = 0
polyknights[bilateral][3] = 4
polyknights[bilateral][4] = 2
polyknights[bilateral][5] = 30
polyknights[bilateral][6] = 32
polyknights[bilateral][7] = 274
polyknights[bilateral][8] = 403
polyknights[bilateral][9] = 2694
polyknights[bilateral][10] = 4604
polyknights[bilateral][11] = 27365

Polyknights without bilateral symmetry (A030448):
polyknights[nonbilateral][1] = 0
polyknights[nonbilateral][2] = 1
polyknights[nonbilateral][3] = 2
polyknights[nonbilateral][4] = 33
polyknights[nonbilateral][5] = 260
polyknights[nonbilateral][6] = 2648
polyknights[nonbilateral][7] = 26105
polyknights[nonbilateral][8] = 267195
polyknights[nonbilateral][9] = 2755322
polyknights[nonbilateral][10] = 28744852
polyknights[nonbilateral][11] = 302092789

and for polyzebras (I haven't checked whether any of these sequences are
already listed):

polyzebras[fixed][1] = 1
polyzebras[fixed][2] = 4
polyzebras[fixed][3] = 28
polyzebras[fixed][4] = 234
polyzebras[fixed][5] = 2162
polyzebras[fixed][6] = 21272
polyzebras[fixed][7] = 218740
polyzebras[fixed][8] = 2323730
polyzebras[fixed][9] = 25314097
polyzebras[fixed][10] = 281299736
polyzebras[fixed][11] = 3176220308

polyzebras[free][1] = 1
polyzebras[free][2] = 1
polyzebras[free][3] = 6
polyzebras[free][4] = 35
polyzebras[free][5] = 290
polyzebras[free][6] = 2714
polyzebras[free][7] = 27524
polyzebras[free][8] = 291073
polyzebras[free][9] = 3166124
polyzebras[free][10] = 35169275
polyzebras[free][11] = 397047529

polyzebras[one-sided][1] = 1
polyzebras[one-sided][2] = 2
polyzebras[one-sided][3] = 8
polyzebras[one-sided][4] = 68
polyzebras[one-sided][5] = 550
polyzebras[one-sided][6] = 5404
polyzebras[one-sided][7] = 54768
polyzebras[one-sided][8] = 581868
polyzebras[one-sided][9] = 6329407
polyzebras[one-sided][10] = 70335294
polyzebras[one-sided][11] = 794064734

polyzebras[bilateral][1] = 1
polyzebras[bilateral][2] = 0
polyzebras[bilateral][3] = 4
polyzebras[bilateral][4] = 2
polyzebras[bilateral][5] = 30
polyzebras[bilateral][6] = 24
polyzebras[bilateral][7] = 280
polyzebras[bilateral][8] = 278
polyzebras[bilateral][9] = 2841
polyzebras[bilateral][10] = 3256
polyzebras[bilateral][11] = 30324

polyzebras[nonbilateral][1] = 0
polyzebras[nonbilateral][2] = 1
polyzebras[nonbilateral][3] = 2
polyzebras[nonbilateral][4] = 33
polyzebras[nonbilateral][5] = 260
polyzebras[nonbilateral][6] = 2690
polyzebras[nonbilateral][7] = 27244
polyzebras[nonbilateral][8] = 290795
polyzebras[nonbilateral][9] = 3163283
polyzebras[nonbilateral][10] = 35166019
polyzebras[nonbilateral][11] = 397017205

-- 
Joseph S. Myers
jsm at polyomino.org.uk