A343095 -id:A343095 - cp's OEIS frontend

A006528 a(n) = (n^4 + n^2 + 2*n)/4.

0, 1, 6, 24, 70, 165, 336, 616, 1044, 1665, 2530, 3696, 5226, 7189, 9660, 12720, 16456, 20961, 26334, 32680, 40110, 48741, 58696, 70104, 83100, 97825, 114426, 133056, 153874, 177045, 202740, 231136, 262416, 296769, 334390, 375480, 420246, 468901, 521664, 578760

Offset: 0

N. J. A. Sloane

Number of ways to color vertices of a square using <= n colors, allowing only rotations.

Also product of first and last terms in n-th row of a triangle of form: row(1)= 1; row(2)= 2,3; row(3) = 4, 5, 6, ... . - Dave Durgin, Aug 17 2012

Nick Baxter, The Burnside di-lemma: combinatorics and puzzle symmetry, in Tribute to a Mathemagician, Peters, 2005, pp. 199-210.
M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row n=2 of A343095.

Cf. A002817 (square colorings).

I:=[0, 1, 6, 24, 70]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 30 2012

A006528:=-z*(1+z+4*z**2)/(z-1)**5; # Simon Plouffe in his 1992 dissertation
a:=n->add(n+add(binomial(n,2), j=1..n),j=0..n):seq(a(n)/2, n=0..35); # Zerinvary Lajos, Aug 26 2008

Table[CycleIndex[CyclicGroup[4],t]/.Table[t[i]->n,{i,1,4}],{n,0,20}] (* Geoffrey Critzer, Mar 13 2011*)
Table[(n^4+n^2+2*n)/4,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,6,24,70},40] (* Harvey P. Dale, Jan 13 2019 *)

a(n) = n*(n+1)*(n^2-n+2)/4; /* Joerg Arndt, Apr 30 2012 */

a(n) = n*(n+1)*(n^2-n+2)/4.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Apr 30 2012
From Stefano Spezia, Oct 11 2023: (Start)
O.g.f.:  x*(1 + x + 4*x^2)/(1 - x)^5.
E.g.f.: exp(x)*x*(4 + 8*x + 6*x^2 + x^3)/4. (End)

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.

Original entry on oeis.org

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.

A047937 Number of 2-colorings of an n X n grid, up to rotational symmetry.

Original entry on oeis.org

1, 2, 6, 140, 16456, 8390720, 17179934976, 140737496748032, 4611686019501162496, 604462909807864344215552, 316912650057057631849169289216, 664613997892457937028364283517337600, 5575186299632655785385110159782842147536896, 187072209578355573530071668259090783437390809661440

Offset: 0

Rob Pratt

Cycle index = 1/4(s_1^(n^2)+ 2 s_4^floor(n^2/4)s_1^(n mod 2)+s_2^floor(n^2/2)s_1^(n mod 2)). - Geoffrey Critzer, Oct 28 2011

			a(2)=6 from
00 10 11 10 11 11
00 00 00 01 10 11

Andrew Howroyd, Table of n, a(n) for n = 0..50
Peter Kagey, Illustration of a(3)=140
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Column k=2 of A343095.
Other columns are A047938, A047939, A047940, A047941, A047942, A047943, A047944, A047945.
Cf. A054247.

Table[(2^(n^2)+2*2^Floor[n^2/4]*2^Mod[n,2]+2^Floor[n^2/2]*2^Mod[n,2])/4,{n,0,10}]  (* Geoffrey Critzer, Oct 28 2011 *)

a(n) = (m^(n^2) + 2*m^((n^2 + 3*(n mod 2))/4) + m^((n^2 + (n mod 2))/2))/4, with m = 2.

Terms a(12) and beyond from Andrew Howroyd, Apr 14 2021

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

A283027 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to rotations.

Original entry on oeis.org

0, 1, 16456, 10763361, 1073758336, 38147070625, 705277897416, 8308234084801, 70368748374016, 463255057977921, 2500000025005000, 11487432519490081, 46221064831264896, 166354152499741921, 544488334821309256, 1642102089568970625, 4611686019501162496

Offset: 0

David Nacin, Feb 27 2017

Cycle index of cyclic group C4 acting on the 16 entries is (2s(4)^4 + s(2)^8 + s(1)^16)/24.

			For n=2 we get a(2)=16456 inequivalent 4 X 4 binary matrices up to rotations.

G. C. Greubel, 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 A343095.
Cf. A282612, A282614, A283026, A283028, A283029, A283030, A283032, A283033.
Cf. A283031 (5 X 5 version), A282613 (3 X 3 version), A006528 (2 X 2 version).

List([0..20], n -> n^4*(n^4+1)*(n^8-n^4+2)/4); # G. C. Greubel, Dec 07 2018

[n^4*(n^4+1)*(n^8-n^4+2)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[n^4*(n^4+1)*(n^8-n^4+2)/4$n=0..18]; # Muniru A Asiru, Dec 07 2018

Table[n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4,{n,0,30}]

a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4; \\ Indranil Ghosh, Feb 27 2017

def A283027(n) : return n**4*(n**4 + 1)*(n**8 - n**4 + 2)/4 # Indranil Ghosh, Feb 27 2017

def A283027(n): return n**4*(n**4*(n**8+1)+2)>>2 # Chai Wah Wu, Jan 24 2023

[n^4*(n^4+1)*(n^8-n^4+2)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.
G.f.: -x*(x + 1)*(x^14 + 16438*x^13 + 10467307*x^12 + 882551228*x^11 + 20463257081*x^10 + 175065647018*x^9 + 626805100059*x^8 + 968894657736*x^7 + 626805100059*x^6 + 175065647018*x^5 + 20463257081*x^4 + 882551228*x^3 + 10467307*x^2 + 16438*x + 1)/(x - 1)^17. (End)
E.g.f.: (1/4)*x*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764054*x^7 + 3281882632*x^6 + 2734926824*x^5 + 1096191600*x^4 + 171800604*x^3 + 7142664*x^2 + 32908*x + 4)*exp(x). - Stefano Spezia, Dec 07 2018
a(n) = n^4*(n^12 + n^4 + 2)/4. - Chai Wah Wu, Jan 24 2023

A283031 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations.

Original entry on oeis.org

0, 1, 8390720, 211822552035, 281474993496064, 74505806274453125, 7107572010747738816, 335267154940213889575, 9444732965876730429440, 179474496923598616041129, 2500000000002500005000000, 27086764858479561198237131, 238490541610199280719585280

Offset: 0

David Nacin, Feb 27 2017

Cycle index of cyclic group C4 acting on the set of 25 entries is (2*s(4)^6*s(1) + s(2)^12*s(1) + s(1)^25).

			For n=2 we get a(2)=8390720 inequivalent 5 X 5 binary matrices up to rotations.

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

Row n=5 of A343095.
Cf. A282612, A282614, A283028, A283029, A283030, A283032, A283033.
Cf. A283027 (4 X 4 version), A282613 (3 X 3 version), A006528 (2 X 2 version).

List([0..30], n -> n^7*(n^2+1)*(n^4-n^2+1)*(n^12-n^6+2)/4); # G. C. Greubel, Dec 07 2018

[n^7*(n^2+1)*(n^4-n^2+1)*(n^12-n^6+2)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[n^7*(n^2+1)*(n^4-n^2+1)*(n^12-n^6+2)/4$n=0..16]; # Muniru A Asiru, Dec 07 2018

Table[n^7(n^2 + 1)(n^4 - n^2 + 1)(n^12 - n^6 + 2)/4, {n, 0, 16}]

a(n) = n^7*(n^2 + 1)*(n^4 - n^2 + 1)*(n^12 - n^6 + 2)/4; \\ Indranil Ghosh, Feb 27 2017

def A283031(n): return n**7*(n**2 + 1)*(n**4 - n**2 + 1)*(n**12 - n**6 + 2)/4 # Indranil Ghosh, Feb 27 2017

def A283031(n): return n**7*(n**6*(n**12+1)+2)>>2 # Chai Wah Wu, Jan 24 2023

[n^7*(n^2+1)*(n^4-n^2+1)*(n^12-n^6+2)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = n^7*(n^2 + 1)*(n^4 - n^2 + 1)*(n^12 - n^6 + 2)/4.
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 + 8390694*x^23 + 211604393640*x^22 + 275970334124554*x^21 + 67256276957109786*x^20 + 5261349807304085586*x^19 + 173956000912091771464*x^18 + 2848226864007694392990*x^17 + 25488973079546662159119*x^16 + 132928565342248912495516*x^15 + 421347175220529448574736*x^14 + 833653141283634765151044*x^13 + 1044911777486454930701740*x^12 + 833653141283634765151044*x^11 + 421347175220529448574736*x^10 + 132928565342248912495516*x^9 + 25488973079546662159119*x^8 + 2848226864007694392990*x^7 + 173956000912091771464*x^6 + 5261349807304085586*x^5 + 67256276957109786*x^4 + 275970334124554*x^3 + 211604393640*x^2 + 8390694*x + 1)/(x - 1)^26. (End)
a(n) = n^7*(n^18 + n^6 + 2)/4. - Chai Wah Wu, Jan 24 2023

A343874 Array read by antidiagonals: T(n,k) is the number of n X n nonnegative integer matrices with sum of elements equal to k, up to rotational symmetry.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 3, 1, 0, 1, 5, 13, 4, 1, 0, 1, 10, 43, 36, 7, 1, 0, 1, 14, 129, 204, 85, 9, 1, 0, 1, 22, 327, 980, 735, 171, 13, 1, 0, 1, 30, 761, 3876, 5145, 2109, 313, 16, 1, 0, 1, 43, 1619, 13596, 29715, 20610, 5213, 528, 21, 1

Offset: 0

Andrew Howroyd, May 06 2021

			Array begins:
=====================================================
n\k | 0  1   2    3     4      5       6        7
----+------------------------------------------------
  0 | 1  0   0    0     0      0       0        0 ...
  1 | 1  1   1    1     1      1       1        1 ...
  2 | 1  1   3    5    10     14      22       30 ...
  3 | 1  3  13   43   129    327     761     1619 ...
  4 | 1  4  36  204   980   3876   13596    42636 ...
  5 | 1  7  85  735  5145  29715  148561   657511 ...
  6 | 1  9 171 2109 20610 164502 1124382  6744582 ...
  7 | 1 13 313 5213 67769 717509 6457529 50732669 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=0..4 are A000007, A000012, A008610, A054771, A054773.
Columns k=0..1 are A000012, A004652.
Cf. A054772 (binary case), A318795, A343095, A343875.

U(n,s)={(s(1)^(n^2) + s(1)^(n%2)*(2*s(4)^(n^2\4) + s(2)^(n^2\2)))/4}
T(n,k)={polcoef(U(n,i->1/(1-x^i) + O(x*x^k)), k)}

A047938 Number of ways to label points of an n X n grid with 3 colors, up to rotational symmetry.

Original entry on oeis.org

1, 3, 24, 4995, 10763361, 211822552035, 37523658921114744, 59824832307866205347043, 858420955073128584419531008641, 110856622060759442496180656754310346403, 128844380183002832759115461915902241562318377784, 1347757724935823407809884872163997148505019182125807296675

Offset: 0

Rob Pratt

nonn

Labels of adjacent points need not be distinct.

Andrew Howroyd, Table of n, a(n) for n = 0..25

Column k=3 of A343095.
See A068253 for number of proper colorings.
Cf. A054739, A047937, A047939.

a(n) = (m^(n^2) + 2 m^((n^2 + 3 (n mod 2))/4) + m^((n^2 + (n mod 2))/2))/4, with m = 3.

Terms a(10) and beyond from Andrew Howroyd, Apr 14 2021

A047939 Number of 4-colorings of an n X n grid, up to rotational symmetry.

Original entry on oeis.org

1, 4, 70, 65824, 1073758336, 281474993496064, 1180591620734591303680, 79228162514264619068554215424, 85070591730234615870455337878516924416, 1461501637330902918203686041642102636484130504704, 401734511064747568885490523085607563280607806359022338048000

Offset: 0

Rob Pratt

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..25

Column k=4 of A343095.

Cf. A054751, A047937.

a(n) = (m^(n^2) + 2 m^((n^2 + 3 (n mod 2))/4) + m^((n^2 + (n mod 2))/2))/4, with m = 4.

Terms a(10) and beyond from Andrew Howroyd, Apr 14 2021

A047940 Number of 5-colorings of an n X n grid, up to rotational symmetry.

Original entry on oeis.org

1, 5, 165, 489125, 38147070625, 74505806274453125, 3637978807092666626953125, 4440892098500626236200333251953125, 135525271560688054250937420874834136962890625, 103397576569128459358926086520114040467888355255126953125

Offset: 0

Rob Pratt

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..25

Column k=5 of A343095.

Cf. A054752, A047937.

a(n) = (m^(n^2) + 2 m^((n^2 + 3 (n mod 2))/4) + m^((n^2 + (n mod 2))/2))/4, with m = 5.

Terms a(9) and beyond from Andrew Howroyd, Apr 14 2021

cp's OEIS Frontend

A006528 a(n) = (n^4 + n^2 + 2*n)/4.

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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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A047937 Number of 2-colorings of an n X n grid, up to rotational symmetry.

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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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A283027 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to rotations.

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A283031 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations.

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