A217331 -id:A217331 - cp's OEIS frontend

A343097 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 21, 102, 1, 0, 1, 5, 55, 2862, 8548, 1, 0, 1, 6, 120, 34960, 5398083, 4211744, 1, 0, 1, 7, 231, 252375, 537157696, 105918450471, 8590557312, 1, 0, 1, 8, 406, 1284066, 19076074375, 140738033618944, 18761832172500795, 70368882591744, 1, 0

Offset: 0

Andrew Howroyd, Apr 14 2021

			Array begins:
====================================================================
n\k | 0 1       2            3               4                 5
----+---------------------------------------------------------------
  0 | 1 1       1            1               1                 1 ...
  1 | 0 1       2            3               4                 5 ...
  2 | 0 1       6           21              55               120 ...
  3 | 0 1     102         2862           34960            252375 ...
  4 | 0 1    8548      5398083       537157696       19076074375 ...
  5 | 0 1 4211744 105918450471 140738033618944 37252918396015625 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..860

Rows 0..5 are A000012, A001477, A002817, A217331, A217338, A283033.
Columns 0..10 are A000007, A000012, A054247, A054739, A054751, A054752, A286392, A286393, A286394, A286396, A286397.
Main diagonal is A001326.
Cf. A246106, A343095, A343875.

T(n,k) = {(k^(n^2) + 2*k^((n^2 + 3*(n%2))/4) + k^((n^2 + (n%2))/2) + 2*k^(n*(n+1)/2) + 2*k^(n*(n+n%2)/2) )/8}

T(n,k) = (k^(n^2) + 2*k^((n^2 + 3*(n mod 2))/4) + k^((n^2 + (n mod 2))/2) + 2*k^(n*(n+1)/2) + 2*k^(n*(n + n mod 2)/2) )/8.

A282613 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to rotations.

Original entry on oeis.org

0, 1, 140, 4995, 65824, 489125, 2521476, 10092775, 33562880, 96870249, 250025500, 589527851, 1290008160, 2651218765, 5165397524, 9611031375, 17180133376, 29647326545, 49590297900, 80672546899, 128000804000, 198571037301, 301818598180, 450289780535

Offset: 0

David Nacin, Feb 19 2017

Cycle index of symmetry group (cyclic rotation group of order 4 acting on the 9 cells of the square) is (2s(4)^2*s(1) + s(2)^4*s(1) + s(1)^9)/4.

			The number of 3 X 3 binary matrices up to rotations is 140.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Row n=3 of A343095.
Cf. A282612, A282614, A217331, A168555.
Cf. A006528 (2 x 2 version), A283027 (4 X 4 version).

Mathematica
```
Table[(2n^3+n^5+n^9)/4, {n, 0, 24}]
```

concat(0, Vec(x*(1 + 130*x + 3640*x^2 + 22054*x^3 + 39070*x^4 + 22054*x^5 + 3640*x^6 + 130*x^7 + x^8) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

a(n) = n^3*(n^2+1)*(n^4-n^2+2)/4.

G.f.: x*(1 + 130*x + 3640*x^2 + 22054*x^3 + 39070*x^4 + 22054*x^5 + 3640*x^6 + 130*x^7 + x^8) / (1 - x)^10. - Colin Barker, Feb 23 2017

A282614 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to vertical and horizontal reflections.

Original entry on oeis.org

0, 1, 168, 5346, 67840, 496875, 2544696, 10151428, 33693696, 97135605, 250525000, 590412966, 1291500288, 2653631071, 5169160920, 9616725000, 17188519936, 29659392873, 49607301096, 80696066410, 128032800000, 198613915731, 301875282808, 450363792396

Offset: 0

David Nacin, Feb 19 2017

Cycle index of symmetry group is (2*s(2)^3*s(1)^3 + s(2)^4*s(1) + s(1)^9)/4.

			The number of 3 X 3 binary matrices up to vertical and horizontal reflections is 168.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A282613, A282614, A217331, A168555. (For 2x2 version see A039623.)

Table[(2n+1+n^4)n^5/4, {n, 0, 24}]
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,1,168,5346,67840,496875,2544696,10151428,33693696,97135605},30] (* Harvey P. Dale, Oct 01 2024 *)

concat(0, Vec(x*(1 + 158*x + 3711*x^2 + 21820*x^3 + 39095*x^4 + 22254*x^5 + 3577*x^6 + 104*x^7) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

a(n) = n^5*(n+1)*(n^3-n^2+n+1)/4.

G.f.: x*(1 + 158*x + 3711*x^2 + 21820*x^3 + 39095*x^4 + 22254*x^5 + 3577*x^6 + 104*x^7) / (1 - x)^10. - Colin Barker, Feb 23 2017

A282612 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to row permutations.

Original entry on oeis.org

0, 1, 120, 3654, 45760, 333375, 1703016, 6784540, 22500864, 64836045, 167167000, 393877506, 861456960, 1769830699, 3447273480, 6412923000, 11461636096, 19776716505, 33076889784, 53804808190, 85365336000, 132422893911, 201268229800, 300266132244, 440396812800

Offset: 0

David Nacin, Feb 19 2017

Cycle index of symmetry group is (3*s(2)^3*s(1)^3 + 2*s(3)^3 + s(1)^9)/6.

			The number of 3 X 3 binary matrices up to row permutations is 120.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A282613, A282614, A217331, A168555. A037270 (2x2 version.)

Mathematica
```
Table[(3n^6+2n^3+n^9)/6, {n, 0, 24}]
```

concat(0, Vec(x*(1 + 110*x + 2499*x^2 + 14500*x^3 + 26015*x^4 + 14934*x^5 + 2365*x^6 + 56*x^7) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

a(n) = n^3*(n^3+2)*(n+1)*(n^2-n+1)/6.

G.f.: x*(1 + 110*x + 2499*x^2 + 14500*x^3 + 26015*x^4 + 14934*x^5 + 2365*x^6 + 56*x^7) / (1 - x)^10. - Colin Barker, Feb 23 2017

A283033 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations and reflections.

Original entry on oeis.org

0, 1, 4211744, 105918450471, 140738033618944, 37252918396015625, 3553786240466361696, 167633579843887699759, 4722366500530551259136, 89737248564744874067889, 1250000000501250002500000, 13543382431328404683826391, 119245270812803151147085824

Offset: 0

David Nacin, Feb 27 2017

nonn

Cycle index of dihedral group D4 acting on the 25 entries is (2*s(4)^6*s(1) + s(2)^{12}*s(1) + 4*s(2)^10*s(1)^5 + s(1)^25)/8.

			For n=2 we get a(2)=4211744 inequivalent 5 X 5 binary matrices up to rotations and reflections.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Row n=5 of A343097.
Cf. A282612, A282613, A282614, A283026, A283027, A283028, A283029, A283030, A283031, A283032.
Cf. A217338 (4 X 4 version), A217331 (3 X 3 version), A002817 (2 X 2 version).

List([0..20], n -> n^7*(n^18+4*n^8+n^6+2)/8); # G. C. Greubel, Dec 07 2018

[n^7*(n^18+4*n^8+n^6+2)/8: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[n^7*(n^18+4*n^8+n^6+2)/8$n=0..16]; # Muniru A Asiru, Dec 07 2018

Table[n^7 (n^18 + 4 n^8 + n^6 + 2)/8, {n, 0, 16}]

a(n) = n^7*(n^18 + 4*n^8 + n^6 + 2)/8; \\ Indranil Ghosh, Feb 27 2017

def A283033(n): return n**7*(n**18 + 4*n**8 + n**6 + 2)/8 # Indranil Ghosh, Feb 27 2017

[n^7*(n^18+4*n^8+n^6+2)/8 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = n^7*(n^18 + 4*n^8 + n^6 + 2)/8.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(x^24 + 4211718*x^23 + 105808945452*x^22 + 137985522720898*x^21 + 33628142067806706*x^20 + 2630674898090394666*x^19 + 86978000386844370748*x^18 + 1424113432167998385342*x^17 + 12744486540004851097263*x^16 + 66464282669989885009756*x^15 + 210673587611186802329496*x^14 + 416826570643036689533748*x^13 + 522455888740564118388412*x^12 + 416826570643036689533748*x^11 + 210673587611186802329496*x^10 + 66464282669989885009756*x^9 + 12744486540004851097263*x^8 + 1424113432167998385342*x^7 + 86978000386844370748*x^6 + 2630674898090394666*x^5 + 33628142067806706*x^4 + 137985522720898*x^3 + 105808945452*x^2 + 4211718*x + 1)/(x - 1)^26. (End)

A217338 Number of inequivalent ways to color a 4 X 4 checkerboard using at most n colors allowing rotations and reflections.

Original entry on oeis.org

0, 1, 8548, 5398083, 537157696, 19076074375, 352654485156, 4154189102413, 35184646816768, 231628411446741, 1250002537502500, 5743722797690911, 23110548002468928, 83177110918426603, 272244240093265636, 821051189587805625, 2305843285702230016, 6082649491072763593

Offset: 0

Geoffrey Critzer, Oct 01 2012

Cycle index of symmetry group: (s(1)^16 + 2*s(4)^4 + 3*s(2)^8 + 2*s(2)^6*s(1)^4)/8.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Row n=4 of A343097.

Cf. A002817, A217331.

Table[(n^16+2n^4+3n^8+2n^10)/8, {n,0,20}]

a(n) = (n^16 + 2*n^4 + 3*n^8 + 2*n^10)/8; \\ Indranil Ghosh, Feb 27 2017

def A217338(n): return (n**16 + 2*n**4 + 3*n**8 + 2*n**10)/8 # Indranil Ghosh, Feb 27 2017

a(n) = (n^16 + 2*n^4 + 3*n^8 + 2*n^10)/8.

G.f.: -x*(x +1)*(x^14 +8530*x^13 +5244373*x^12 +441307760*x^11 +10231414811*x^10 +87532894238*x^9 +313403397135*x^8 +484445834304*x^7 +313403397135*x^6 +87532894238*x^5 +10231414811*x^4 +441307760*x^3 +5244373*x^2 +8530*x +1)/(x -1)^17. [Colin Barker, Oct 04 2012]

cp's OEIS Frontend

A343097 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.

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A282613 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to rotations.

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A282614 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to vertical and horizontal reflections.

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A282612 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to row permutations.

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A283033 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations and reflections.

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A217338 Number of inequivalent ways to color a 4 X 4 checkerboard using at most n colors allowing rotations and reflections.

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