A000315 -id:A000315 - 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

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

A003090 Number of species (or "main classes" or "paratopy classes") of Latin squares of order n.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 147, 283657, 19270853541, 34817397894749939, 2036029552582883134196099

Offset: 1

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 231.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Yue Guan, Minjia Shi, Denis S. Krotov, The Steiner triple systems of order 21 with a transversal subdesign TD(3,6), arXiv:1905.09081 [math.CO], 2019.
A. Hulpke, P. Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219
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.
M. G. Palomo, Latin polytopes, arXiv preprint arXiv:1402.0772 [math.CO], 2014-2016.
Giancarlo Urzua, On line arrangements with applications to 3-nets, arXiv:0704.0469 [math.AG], 2007-2009 (see page 9).
Ian M. Wanless, A Generalization of Transversals for Latin Squares, Electronic Journal of Combinatorics, volume 9, number 1 (2002), R12.
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

Cf. A000315, A002860, A040082.

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

A057771 Number of loops (quasigroups with an identity element) of order n.

Original entry on oeis.org

0, 1, 1, 1, 2, 6, 109, 23746, 106228849, 9365022303540, 20890436195945769617, 1478157455158044452849321016

Offset: 0

Christian G. Bower, Nov 01 2000

Brendan D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs 15 (2007), no. 2, 98-119.
A. Hulpke, Petteri Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219.
Index entries for sequences related to quasigroups

Cf. A000315, A057991, A057992, A057993, A057994, A057996, A057997, A057998, A089925.

a(8) from Juergen Buntrock (jubu(AT)jubu.com), Nov 03 2003.
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
a(0) prepended by Jianing Song, Oct 26 2019

A274171 Number of diagonal Latin squares of order n with the first row in order.

Original entry on oeis.org

1, 0, 0, 2, 8, 128, 171200, 7447587840, 5056994653507584

Offset: 1

Eduard I. Vatutin, Jul 07 2016

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Sep 29 2020

			The a(4) = 2 diagonal Latin squares are:
   0 1 2 3   0 1 2 3
   2 3 0 1   3 2 1 0
   3 2 1 0   1 0 3 2
   1 0 3 2   2 3 0 1
.
The a(5) = 8 diagonal Latin squares are:
   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
   1 3 4 2 0   1 4 3 0 2   2 3 4 0 1   2 4 1 0 3
   4 2 1 0 3   3 2 1 4 0   4 0 1 2 3   4 0 3 2 1
   2 0 3 4 1   4 3 0 2 1   1 2 3 4 0   3 2 4 1 0
   3 4 0 1 2   2 0 4 1 3   3 4 0 1 2   1 3 0 4 2
.
   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
   3 4 0 1 2   3 4 1 2 0   4 2 0 1 3   4 2 3 0 1
   1 2 3 4 0   4 2 3 0 1   1 4 3 2 0   3 4 1 2 0
   4 0 1 2 3   2 0 4 1 3   3 0 1 4 2   1 3 0 4 2
   2 3 4 0 1   1 3 0 4 2   2 3 4 0 1   2 0 4 1 3

S. E. Kochemazov, E. I. Vatutin, and O. S. Zaikin, Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order, arXiv:1709.02599 [math.CO], 2017.
S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project
Eduard I. Vatutin, a(9) value fixed after
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. 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).
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S.Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.
Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., The effect of filling cells order to the rate of generation of diagonal Latin squares, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39 (in Russian).
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).
Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., and Titov V.S., 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.
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)
Index entries for sequences related to Latin squares and rectangles

Cf. A000315, A000479, A274806, A287764, A309283.

a(n) = A274806(n)/n!.

a(9) added from Vatutin et al. (2016) by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

A274806 Number of diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 48, 960, 92160, 862848000, 300286741708800, 1835082219864832081920

Offset: 1

Eduard I. Vatutin, Jul 07 2016

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Oct 05 2020

S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
M. O. Manzuk and N. N. Nikitina, About the number of diagonal Latin squares of order 9 as a one of results of RakeSearch distributed computing project
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. 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).
E. I. Vatutin, O. S. Zaikin, A. D. Zhuravlev, M. O. Manzuk, S. E. Kochemazov and V. S. Titov, Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares, Proceedings of Distributed Computing and grid-technologies in science and education (GRID'16), JINR, Dubna, 2016, pp. 114-115.
E. I. Vatutin, S. E. Kochemazov, and O. S. Zaikin, Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9, Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol. 753, pp. 114-129. doi: 10.1007/978-3-319-67035-5_9.
Eduard I. Vatutin, Stepan E. Kochemazov, Oleq S. Zaikin, Maxim O. Manzuk, Natalia N. Nikitina, and Vitaly S. Titov, Central symmetry properties for diagonal Latin squares, Problems of Information Technology (2019) No. 2, 3-8.
Vatutin E. I., Zaikin O. S., Zhuravlev A. D., Manzuk M. O., Kochemazov S. E., and Titov V. S., The effect of filling cells order to the rate of generation of diagonal Latin squares, Information-measuring and diagnosing control systems (Diagnostics - 2016). Kursk: SWSU, 2016. pp. 33-39, (in Russian).
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).
Eduard I. Vatutin, a(9) value fixed
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.
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)
Index entries for sequences related to Latin squares and rectangles

Cf. A000315, A000479, A274171, A287764, A309283.

a(n) = A274171(n) * n!.

a(9) from Vatutin et al. (2016) added by Max Alekseyev, Oct 05 2016

a(9) corrected by Eduard I. Vatutin, Oct 20 2016

A000479 Number of 1-factorizations of K_{n,n}.

Original entry on oeis.org

1, 1, 1, 2, 24, 1344, 1128960, 12198297600, 2697818265354240, 15224734061278915461120, 2750892211809148994633229926400, 19464657391668924966616671344752852992000

Offset: 0

N. J. A. Sloane

Also, number of Latin squares of order n with first row 1,2,...,n.
Also number of fixed diagonal Latin squares of order n. - Eric W. Weisstein, Dec 18 2005
Also maximum number of Latin squares of order n such that no two of them have all the same rows (respectively, columns). - Rick L. Shepherd, Mar 01 2008

CRC Handbook of Combinatorial Designs, 1996, p. 660.
Denes and Keedwell, Latin Squares and Applications, Academic Press 1974.

B. D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs 15 (2007), no. 2, 98-119.
B. D. McKay and I. M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.
Artur Schaefer, Endomorphisms of The Hamming Graph and Related Graphs, arXiv preprint arXiv:1602.02186 [math.CO], 2016.
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, 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)
Vatutin E.I., Zaikin O.S., Zhuravlev A.D., Manzyuk M.O., Kochemazov S.E., Titov V.S., 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
Index entries for sequences related to Latin squares and rectangles

Cf. A000315, A000528, A002860.

See A040082 and A264603 for other versions.

a(n) = A000315(n)*(n-1)! = A002860(n)/n!.

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

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

A351409 a(n) = n(n!)^(2n-2).

Original entry on oeis.org

1, 8, 3888, 764411904, 214990848000000000, 224634374557469245440000000000, 1880461634768804771224006806208512000000000000, 240091793104790737576620139562796649430329798636339200000000000000, 813675117804798213250391541747787241264315446434692481270971279693253181440000000000000000

Offset: 1

Dan Eilers, Feb 11 2022

nonn

a(n) is the number of reduced Stable Marriage Problem instances of order n. In the SMP, relabeling men or women has no effect on the number of stable matchings. So the men and women can be relabeled to normalize the order of man #1's rankings (with woman #1 as his first choice and woman n as his last choice), and to similarly normalize the order of woman #1's rankings, except for her ranking of man #1. This reduces the number of possible instances by a factor of n!(n-1)! (A010790 with shifted offset), from (n!)^(2n) (A185141) to a(n). This reduction is directly analogous to the identical reduction from latin squares (A002860) to reduced latin squares (A000315), and can be directly applied to the Latin Stable Marriage Problem (A351413). As with reduced latin squares, some further reduction is possible analogous to row/column reduced latin squares (A123234).
It is tempting to aim for a reduction of (n!)^2 by simultaneously normalizing all of man #1 and woman #1's preferences, but that isn't possible unless man #1 and woman #1 happen to be mutual first choices.
Applying this reduction to A344669 reduces A344669(2) and A344669(4) to 1, demonstrating that these maximal instances arising in A005154 are unique up to participant relabeling. It raises the question of which other values of n make A344669(n) reducible to 1.

Andrew Howroyd, Table of n, a(n) for n = 1..20
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, [Section 7, Symmetries].

Cf. A185141, A000315, A351413, A344669.

a[n_] := n*(n!)^(2*n - 2); Table[a[n], {n, 1, 9}] (* Robert P. P. McKone, Feb 12 2022 *)

a(n) = n*(n!)^(2*n-2) \\ Andrew Howroyd, Feb 12 2022

a(n) = A185141(n) / A010790(n-1).

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

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A003090 Number of species (or "main classes" or "paratopy classes") of Latin squares of order n.

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A057771 Number of loops (quasigroups with an identity element) of order n.

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A274171 Number of diagonal Latin squares of order n with the first row in order.

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A274806 Number of diagonal Latin squares of order n.

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A000479 Number of 1-factorizations of K_{n,n}.

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A351409 a(n) = n(n!)^(2n-2).