A003090 -id:A003090 - cp's OEIS frontend

A007283 a(n) = 3*2^n.

3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - Labos Elemer, May 07 2002
Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940. - Reinhard Zumkeller, Feb 25 2003
Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - Benoit Cloitre, Mar 12 2003
The sequence of first differences is this sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005
Subsequence of A122132. - Reinhard Zumkeller, Aug 21 2006
Apart from the first term, a subsequence of A124509. - Reinhard Zumkeller, Nov 04 2006
Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007
Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} -> {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n). - Milan Janjic, May 10 2007
a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953). - Jaroslav Krizek, Aug 17 2009
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012
a(n-1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212. - Jon Perry, Oct 10 2012
If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x-4. This equation also has solutions 84, 3348, 1450092, ...  which are not of the form 3*2^n. - Farideh Firoozbakht, Nov 30 2013
a(n) is the upper bound for the "X-ray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu). - L. Edson Jeffery, Jan 11 2014
If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n-2) many open sets. See Brown and Stephen references. - Ross La Haye, Jan 19 2014
Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)
Fix a dimension n. For a real-valued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all k-point subsets S in E.  Moreover, the best possible k is 3 * 2^(n-1).
The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.
For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)
Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The average of consecutive powers of 2 beginning with 2^1. - Melvin Peralta and Miriam Ong Ante, May 14 2016
For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207). - Ivan N. Ianakiev, Dec 09 2016
Also, for n >= 2, the number of length-n strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit) - Jeffrey Shallit, Dec 02 2019
Also, a(n) is the minimum link-length of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)-dimensional hypercube. - Marco Ripà, Aug 22 2022
The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - Antti Karttunen, Sep 06 2023
The finite subsequence a(3), a(4), a(5), a(6) = 24, 48, 96, 192 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A000244 (see comment there). - Felix Huber, Feb 15 2024
A level 1 Sierpiński triangle is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.  For n>2, a(n-3) is the radius of the level n Sierpiński triangle graph. - Allan Bickle, Aug 03 2024

Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
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
K. Bezdek and Tudor Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Tomislav Došlić, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
John Elias, Illustration: 2^n+1 hexagram perimeters
Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Tanya Khovanova, Recursive Sequences
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
Index entries for linear recurrences with constant coefficients, signature (2).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Subsequence of the following sequences: A029744, A029747, A029748, A029750, A362804 (after 3), A364494, A364496, A364289, A364291, A364292, A364295, A364497, A364964, A365422.
Essentially same as A003945 and A042950.
Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
Cf. A000079, A100540, A124508, A221718.
Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.
Cf. A133466, A225546, A163511, A243071.
Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959.

a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012

[3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011

A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013

Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)

A007283(n):=3*2^n$
makelist(A007283(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

PARI
```
a(n)=3*2^n
```

a(n)=3<Charles R Greathouse IV, Oct 10 2012

def A007283(n): return 3<Chai Wah Wu, Feb 14 2023

(List.fill(40)(2: BigInt)).scanLeft(1: BigInt)( * ).map(3 * ) // _Alonso del Arte, Nov 28 2019

G.f.: 3/(1-2*x).
a(n) = 2*a(n - 1), n > 0; a(0) = 3.
a(n) = Sum_{k = 0..n} (-1)^(k reduced (mod 3))*binomial(n, k). - Benoit Cloitre, Aug 20 2002
a(n) = A118416(n + 1, 2) for n > 1. - Reinhard Zumkeller, Apr 27 2006
a(n) = A000079(n) + A000079(n + 1). - Zerinvary Lajos, May 12 2007
a(n) = A000079(n)*3. - Omar E. Pol, Dec 16 2008
From Paul Curtz, Feb 05 2009: (Start)
a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.
a(n) = abs(b(n) - b(n+3)) with b(n) = (-1)^n*A084247(n). (End)
a(n) = 2^n + 2^(n + 1). - Jaroslav Krizek, Aug 17 2009
a(n) = A173786(n + 1, n) = A173787(n + 2, n). - Reinhard Zumkeller, Feb 28 2010
A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0. - Reinhard Zumkeller, Sep 01 2012
a(n) = (A000225(n) + 1)*3. - Martin Ettl, Nov 11 2012
E.g.f.: 3*exp(2*x). - Ilya Gutkovskiy, May 15 2016
A020651(a(n)) = 2. - Yosu Yurramendi, Jun 01 2016
a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2). - Ezhilarasu Velayutham, Sep 01 2019
a(A048672(n)) = A225546(A133466(n)). - Michel Marcus and Peter Munn, Nov 29 2019
Sum_{n>=1} 1/a(n) = 2/3. - Amiram Eldar, Oct 28 2020

A002860 Number of Latin squares of order n; or labeled quasigroups.

Original entry on oeis.org

1, 2, 12, 576, 161280, 812851200, 61479419904000, 108776032459082956800, 5524751496156892842531225600, 9982437658213039871725064756920320000, 776966836171770144107444346734230682311065600000

Offset: 1

N. J. A. Sloane

Also the number of minimum vertex colorings in the n X n rook graph. - Eric W. Weisstein, Mar 02 2024

David Nacin, "Puzzles, Parity Maps, and Plenty of Solutions", Chapter 15, The Mathematics of Various Entertaining Subjects: Volume 3 (2019), Jennifer Beineke & Jason Rosenhouse, eds. Princeton University Press, Princeton and Oxford, p. 245.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ronald Alter, Research Problems: How Many Latin Squares are There?, Amer. Math. Monthly 82 (1975), no. 6, 632-634. MR1537769.
Stanley E. Bammel and Jerome Rothstein, The number of 9 X 9 Latin squares, Discrete Math., 11 (1975), 93-95.
Daniel Berend, On the number of Sudoku squares, Discrete Mathematics 341.11 (2018): 3241-3248. See p. 3241.
Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, Some Properties of Latin Squares, Laboratorio Emmy Noether, 2009.
Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021.
John W. Brown, Enumeration of Latin squares with application to order 8, J. Combin. Theory, 5 (1968), 177-184.
Nikhil Byrapuram, Hwiseo (Irene) Choi, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Evin Liang, Rajarshi Mandal, Aika Oki, Daniel Wu, and Michael Yang, Quad Squares, arXiv:2308.07455 [math.HO], 2023.
Gloria S. Choi, Beyond Sudoku: New puzzle variants based on a novel mathematical mapping, J. High Sch. Sci. (2025) Vol. 9, No. 2, 70-84.
Hai-Dang Dau and Nicolas Chopin, Waste-free Sequential Monte Carlo, arXiv:2011.02328 [stat.CO], 2020.
Abdelrahman Desoky, Hany Ammar, Gamal Fahmy, Shaker El-Sappagh, Abdeltawab Hendawi, and Sameh H. Basha, Latin Square and Artificial Intelligence Cryptography for Blockchain and Centralized Systems, Int'l Conf. Adv. Intel. Sys. Informat., Proc. 9th Int'l Conf. (AISI 2023) pp. 444-455.
Thangavelu Geetha, Amritanshu Prasad, and Shraddha Srivastava, Schur Algebras for the Alternating Group and Koszul Duality, arXiv:1902.02465 [math.RT], 2019.
E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189-198. MR0179095 (31 #3346). Mentions this sequence. - _N. J. A. Sloane_, Mar 15 2014
Yue Guan, Minjia Shi and Denis S. Krotov, The Steiner triple systems of order 21 with a transversal subdesign TD(3,6), arXiv:1905.09081 [math.CO], 2019.
Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, Fun with Latin Squares, arXiv:2109.01530 [math.HO], 2021.
Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
A.-A. A. Jucys, The number of distinct Latin squares as a group-theoretical constant, J. Combinatorial Theory Ser. A 20 (1976), no. 3, 265-272. MR0419259 (54 #7283).
Dieter Jungnickel and Vladimir D. Tonchev, Counting Steiner triple systems with classical parameters and prescribed rank, arXiv:1709.06044 [math.CO], 2017.
Lintao Liu, Xuehu Yan, Yuliang Lu, and Huaixi Wang, 2-threshold Ideal Secret Sharing Schemes Can Be Uniquely Modeled by Latin Squares, National University of Defense Technology, Hefei, China, (2019).
Brendan D. McKay, Alison Meynert and Wendy Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Des. 15 (2007), no. 2, 98-119.
Brendan D. McKay and Eric Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
Brendan D. McKay and Ian M. Wanless, On the number of Latin squares. Preprint 2004.
Brendan D. McKay and Ian M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.
Jia-yu Shao and Wan-di Wei, A formula for the number of Latin squares., Discrete Mathematics 110 (1992) 293-296.
Minjia Shi, Li Xu, and Denis S. Krotov, The number of the non-full-rank Steiner triple systems, arXiv:1806.00009 [math.CO], 2018.
Douglas S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
Douglas S. Stones and Ian M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.
Eric Weisstein's World of Mathematics, Latin Square.
Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
Eric Weisstein's World of Mathematics, Rook Graph.
Mark B. Wells, The number of Latin squares of order 8, J. Combin. Theory, 3 (1967), 98-99.
Krasimir Yordzhev, The bitwise operations in relation to obtaining Latin squares, arXiv preprint arXiv:1605.07171 [cs.OH], 2016.
Index entries for sequences related to Latin squares and rectangles
Index entries for sequences related to quasigroups

Cf. A000315, A000479.
Cf. A003090, A040082, A057991.
Cf. A098679 (Latin cubes).
A row of the array in A249026.

Table[Length[ResourceFunction["FindProperColorings"][GraphProduct[CompleteGraph[n], CompleteGraph[n], "Cartesian"], n]], {n, 5}]

a(n) = n!*A000479(n) = n!*(n-1)!*A000315(n).

One more term (from the McKay-Wanless article) from Richard Bean, Feb 17 2004

A000315 Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.

Original entry on oeis.org

1, 1, 1, 4, 56, 9408, 16942080, 535281401856, 377597570964258816, 7580721483160132811489280, 5363937773277371298119673540771840

Offset: 1

N. J. A. Sloane

A reduced Latin square of order n is an n X n matrix where each row and column is a permutation of 1..n and the first row and column are 1..n in increasing order. - Michael Somos, Mar 12 2011

The Stones-Wanless (2010) paper shows among other things that a(n) is 0 mod n if n is composite and 1 mod n if n is prime.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 183.
J. Denes, A. D. Keedwell, editors, Latin Squares: new developments in the theory and applications, Elsevier, 1991, pp. 1, 388.
R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research. 6th ed., Hafner, NY, 1963, p. 22.
C. R. Rao, S. K. Mitra and A. Matthai, editors, Formulae and Tables for Statistical Work. Statistical Publishing Society, Calcutta, India, 1966, p. 193.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, pp. 37, 53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 240.

S. E. Bammel and J. Rothstein, The number of 9x9 Latin squares, Discrete Math., 11 (1975), 93-95.
Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, Some Properties of Latin Squares, Laboratorio Emmy Noether, 2009.
Nikhil Byrapuram, Hwiseo (Irene) Choi, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Evin Liang, Rajarshi Mandal, Aika Oki, Daniel Wu, and Michael Yang, Quad Squares, arXiv:2308.07455 [math.HO], 2023.
B. Cherowitzo, Latin Squares, Comb. Structures Lecture Notes.
Gheorghe Coserea, Solutions for n=5.
Gheorghe Coserea, Solutions for n=6.
Gheorghe Coserea, MiniZinc model for generating solutions.
E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189--198. MR0179095 (31 #3346). Mentions this sequence. - _N. J. A. Sloane_, Mar 15 2014
Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 1, Article #S1H1.
B. D. McKay, A. Meynert and W. Myrvold, Small latin squares, quasigroups and loops, J. Combin. Designs, vol. 15, no. 2 (2007) pp. 98-119.
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
B. D. McKay and I. M. Wanless, On the number of Latin squares. Preprint 2004.
B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.
Young-Sik Moon, Jong-Yoon Yoon, Jong-Seon No, and Sang-Hyo Kim, Interference Alignment Schemes Using Latin Square for Kx3 MIMO X Channel, arXiv:1810.05400 [cs.IT], 2018.
Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares, arXiv:2103.11018 [cs.DM], 2021. Mentions this sequence.
J. Shao and W. Wei, A formula for the number of Latin squares., Discrete Mathematics 110 (1992) 293-296.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
D. S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.
E. I. Vatutin, V. S. Titov, O. S. Zaikin, S. E. Kochemazov, S. U. Valyaev, A. D. Zhuravlev, and M. O. Manzuk, Using grid systems for enumerating combinatorial objects with example of diagonal Latin squares, Information technologies and mathematical modeling of systems (2016), pp. 154-157. (in Russian)
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzyuk, S. E. Kochemazov, and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares. CEUR Workshop proceedings. Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education. 2017. Vol. 1787. pp. 486-490. urn:nbn:de:0074-1787-5.
Eric Weisstein's World of Mathematics, Latin Square.
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Index entries for sequences related to Latin squares and rectangles
Index entries for sequences related to quasigroups

Cf. A000479, A002860, A003090, A040082, A057771, A057997.

a(n) = A002860(n) / (n! * (n-1)!) = A000479(n) / (n-1)!.

Added June 1995: the 10th term was probably first computed by Eric Rogoyski

a(11) (from the McKay-Wanless article) from Richard Bean, Feb 17 2004

A040082 Number of inequivalent Latin squares (or isotopy classes of Latin squares) of order n.

Original entry on oeis.org

1, 1, 1, 2, 2, 22, 564, 1676267, 115618721533, 208904371354363006, 12216177315369229261482540

Offset: 1

N. J. A. Sloane

Here "isotopy class" means an equivalence class of Latin squares under the operations of row permutation, column permutation and symbol permutation. [Brendan McKay]

R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and Medical Research. 6th ed., Hafner, NY, 1963, p. 22.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. W. Brown, Enumeration of Latin squares with application to order 8, J. Combin. Theory, 5 (1968), 177-184. [a(7) and a(8) appear to be given incorrectly. - _N. J. A. Sloane_, Jan 23 2020]
A. Hulpke, Petteri Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219.
G. Kolesova, C. W. H. Lam and L. Thiel, On the number of 8x8 Latin squares, J. Combin. Theory,(A) 54 (1990) 143-148.
Brendan D. McKay, Latin Squares (has list of all such squares)
Brendan D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs 15 (2007), no. 2, 98-119.
Brendan D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
Eduard Vatutin, Alexey Belyshev, Stepan Kochemazov, Oleg Zaikin, Natalia Nikitina, Enumeration of Isotopy Classes of Diagonal Latin Squares of Small Order Using Volunteer Computing, Russian Supercomputing Days (Суперкомпьютерные дни в России), 2018.
Eric Weisstein's World of Mathematics, Latin Square
M. B. Wells, The number of Latin squares of order 8, J. Combin. Theory, 3 (1967), 98-99.
Index entries for sequences related to Latin squares and rectangles

Cf. A002860, A003090, A000315. See A000528 for another version.

7 X 7 and 8 X 8 results confirmed by Brendan McKay
Beware: erroneous versions of this sequence can be found in the literature!
a(9)-a(10) (from the McKay-Meynert-Myrvold article) from Richard Bean, Feb 17 2004
a(11) from Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009

A287764 Number of main classes of diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 972, 4873096, 3292326155394

Offset: 1

Eduard I. Vatutin, May 31 2017

A. D. Belyshev, Discussion about properties of diagonal Latin squares (in Russian)
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
E. I. Vatutin, A. D. Belyshev, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
E. I. Vatutin, Enumerating the diagonal Latin squares of order 8 using equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. I. Vatutin, Enumerating the diagonal Latin squares of order 9 using Gerasim@Home volunteer distributed computing project, equivalence classes of X-based fillings of diagonals and ESODLS-schemas (in Russian)
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Supercomputing Days Russia 2018, Moscow, Moscow State University, 2018, pp. 933-942.
Eduard Vatutin, Alexey Belyshev, Stepan Kochemazov, Oleg Zaikin, Natalia Nikitina, Enumeration of Isotopy Classes of Diagonal Latin Squares of Small Order Using Volunteer Computing, Russian Supercomputing Days (Суперкомпьютерные дни в России), 2018.
E. Vatutin, A. Belyshev, S. Kochemazov, O. Zaikin, N. Nikitina, Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing, Communications in Computer and Information Science. Vol. 965. Springer, 2018. pp. 578-586.
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina, Maxim O. Manzuk, First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch. Third independent confirmation of the previously calculated value a(9) (in Russian).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
Index entries for sequences related to Latin squares and rectangles

Cf. A003090, A309283, A274171.

a(9) from Eduard I. Vatutin, Jul 06 2019

A035481 Number of n X n symmetric matrices whose first row is 1..n and whose rows and columns are all permutations of 1..n.

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 456, 6240, 10936320, 1225566720, 130025295912960, 252282619805368320, 2209617218725251404267520, 98758655816833727741338583040

Offset: 0

Joshua Zucker and Joe Keane

The odd subsequence is A000438. The even subsequence is A035483.

			a(3) = 1 because after 123 in the first row and column, 213 is not allowed for the second row, so it must be 231 and thus the third row is 312.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130, also on Enumeration of Latin squares with conjugate symmetry, arXiv:2104.07902 [math.CO], 2021. Table 2 p. 7.

Cf. A000438, A035482, A000315, A002860, A003090, A040082.

(* This script is not suitable for n > 6 *) matrices[n_ /; n > 1] := Module[{a, t, vars}, t = Table[Which[i==1, j, j==1, i, j>i, a[i, j], True, a[j, i]], {i, n}, {j, n}]; vars = Select[Flatten[t], !IntegerQ[#]& ] // Union; t /. {Reduce[And @@ (1 <= # <= n & /@ vars) && And @@ Unequal @@@ t, vars, Integers] // ToRules}]; a[0] = a[1] = 1; a[n_] := Length[ matrices[n]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 6}] (* Jean-François Alcover, Jan 04 2016 *)

a(10)-a(13) from Ian Wanless, Oct 20 2019

A129732 Number of main classes of Graeco-Latin squares of order n. That is, species of pairs of orthogonal Latin squares.

Original entry on oeis.org

1, 1, 1, 0, 7, 2165, 91846374

Offset: 3

Brendan McKay, May 13 2007

			The unique example for n=3 has two Latin squares [[0, 2, 1], [2, 1, 0], [1, 0, 2]] and [[0, 2, 1], [1, 0, 2], [2, 1, 0]].

J. Egan and I. M. Wanless, Enumeration of MOLS of small order, Mathematics of Computation 85, 2016, 799-824.
B. D. McKay, Latin squares.
I. Wanless, Data on mutually orthogonal Latin squares (MOLS).

Cf. A003090, A266166, A266167, A266168, A266169, A266170, A266171, A266172, A266173.

a(9) by Egan and Wanless added by Janne I. Kokkala, Sep 08 2015

A309344 a(n) is the number of distinct numbers of transversals of order n Latin squares.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 36, 74

Offset: 1

Alvaro R. Belmonte, Eugene Fiorini, Peterson Lenard, Froylan Maldonado, Sabrina Traver, Wing Hong Tony Wong, Jul 24 2019

We found all transversals in the main class Latin square representatives of order n.
These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.
For all spectra of even orders all known values included in them are divisible by 2. For all spectra of orders n=4k+2 all known values included in the corresponding spectra are divisible by 4. - Eduard I. Vatutin, Mar 01 2025
a(9)>=407, a(10)>=463, a(11)>=6437, a(12)>=23715. - Eduard I. Vatutin, added Mar 01 2025, updated Aug 14 2025

			For n=7, the number of transversals that an order 7 Latin square may have is 3, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 41, 43, 45, 47, 55, 63, or 133. Hence there are 36 distinct numbers of transversals of order 7 Latin squares, so a(7)=36.

Brendan McKay, Combinatorial Data.
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, On the number of transversals in diagonal Latin squares of even orders (in Russian), Cloud and distributed computing systems, within the National supercomputing forum (NSCF - 2023). Pereslavl-Zalessky, 2023. pp. 101-105.
Index entries for sequences related to Latin squares and rectangles.

Cf. A003090, A090741 (maximum number), A091323 (minimum number), A301371, A308853, A309088, A344105 (version for diagonal Latin squares).

%This extracts entries from each column.  For an example, if
%A=[1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16], and if list = (2, 1, 4),
%this code extracts the second element in the first column, the first
%element in the second column, and the fourth element in the third column.
function [output] = extract(matrix,list)
for i=1:length(list)
    output(i) = matrix(list(i),i);
end
end
%Searches matrix to find transversal and outputs the transversal.
function [output] = findtransversal(matrix)
n=length(matrix);
for i=1:n
    partialtransversal(i,1)=i;
end
for i=2:n
    newpartialtransversal=[];
    for j=1:length(partialtransversal)
        for k=1:n
            if (~ismember(k,partialtransversal(j,:)))&(~ismember(matrix(k,i),extract(matrix,partialtransversal(j,:))))
                newpartialtransversal=[newpartialtransversal;[partialtransversal(j,:),k]];
            end
        end
    end
    partialtransversal=newpartialtransversal;
end
output=partialtransversal;
end
%Takes input of n^2 numbers with no spaces between them and converts it
%into an n by n matrix.
function [A] = tomatrix(input)
n=sqrt(floor(log10(input))+2);
for i=1:n^2
    temp(i)=mod(floor(input/(10^(i-1))),10);
end
for i=1:n
    for j=1:n
        A(i,j)=temp(n^2+1-(n*(i-1)+j));
    end
end
A=A+ones(n);
end

cp's OEIS Frontend

A007283 a(n) = 3*2^n.

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A002860 Number of Latin squares of order n; or labeled quasigroups.

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A000315 Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.

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A040082 Number of inequivalent Latin squares (or isotopy classes of Latin squares) of order n.

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A287764 Number of main classes of diagonal Latin squares of order n.

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A035481 Number of n X n symmetric matrices whose first row is 1..n and whose rows and columns are all permutations of 1..n.

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A129732 Number of main classes of Graeco-Latin squares of order n. That is, species of pairs of orthogonal Latin squares.

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A309344 a(n) is the number of distinct numbers of transversals of order n Latin squares.

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