Displaying 1-10 of 18 results found.
1, 1, 1, 1, 2, 2, 22, 563, 1676257
1, 2, 3, 5, 7, 29, 593, 1676860, 115620398393, 208904486974761399, 12216177524273716236243939
COMMENTS
Partial sums of number of inequivalent Latin squares (or isotopy classes of Latin squares) of order n. The subsequence of primes (6 in a row) in this partial sum begins: 2, 3, 5, 7, 29, 593.
EXAMPLE
a(7) = 1 + 1 + 1 + 2 + 2 + 22 + 564 = 593 is prime.
a(n) = 3*2^n.
(Formerly M2561)
+10
241
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
COMMENTS
Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
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
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
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 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
REFERENCES
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).
FORMULA
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) = abs(b(n) - b(n+3)) with b(n) = (-1)^n* A084247(n). (End)
PROG
(PARI) a(n)=3*2^n
(Haskell)
a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
(Scala) (List.fill(40)(2: BigInt)).scanLeft(1: BigInt)(_ * _).map(3 * _) // Alonso del Arte, Nov 28 2019
(Python)
CROSSREFS
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.
Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
Number of Latin squares of order n; or labeled quasigroups.
(Formerly M2051 N0812)
+10
39
1, 2, 12, 576, 161280, 812851200, 61479419904000, 108776032459082956800, 5524751496156892842531225600, 9982437658213039871725064756920320000, 776966836171770144107444346734230682311065600000
COMMENTS
Also the number of minimum vertex colorings in the n X n rook graph. - Eric W. Weisstein, Mar 02 2024
REFERENCES
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).
LINKS
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.
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.
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.
Eric Weisstein's World of Mathematics, Rook Graph.
MATHEMATICA
Table[Length[ResourceFunction["FindProperColorings"][GraphProduct[CompleteGraph[n], CompleteGraph[n], "Cartesian"], n]], {n, 5}]
EXTENSIONS
One more term (from the McKay-Wanless article) from Richard Bean, Feb 17 2004
Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.
(Formerly M3690 N1508)
+10
24
1, 1, 1, 4, 56, 9408, 16942080, 535281401856, 377597570964258816, 7580721483160132811489280, 5363937773277371298119673540771840
COMMENTS
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.
REFERENCES
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.
LINKS
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.
Brian Hopkins, Euler's Enumerations, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 1, Article #S1H1.
EXTENSIONS
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
Number of species (or "main classes" or "paratopy classes") of Latin squares of order n.
(Formerly M0387)
+10
13
1, 1, 1, 2, 2, 12, 147, 283657, 19270853541, 34817397894749939, 2036029552582883134196099
REFERENCES
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).
LINKS
M. G. Palomo, Latin polytopes, arXiv preprint arXiv:1402.0772 [math.CO], 2014-2016.
EXTENSIONS
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
Number of 1-factorizations of K_{n,n}.
+10
10
1, 1, 1, 2, 24, 1344, 1128960, 12198297600, 2697818265354240, 15224734061278915461120, 2750892211809148994633229926400, 19464657391668924966616671344752852992000
COMMENTS
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
REFERENCES
CRC Handbook of Combinatorial Designs, 1996, p. 660.
Denes and Keedwell, Latin Squares and Applications, Academic Press 1974.
EXTENSIONS
a(11) (from the McKay-Wanless article) from Richard Bean, Feb 17 2004
Number of n X n symmetric matrices whose first row is 1..n and whose rows and columns are all permutations of 1..n.
+10
10
1, 1, 1, 1, 4, 6, 456, 6240, 10936320, 1225566720, 130025295912960, 252282619805368320, 2209617218725251404267520, 98758655816833727741338583040
EXAMPLE
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.
MATHEMATICA
(* 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(n) is the minimum absolute value of nonzero determinants of order n Latin squares.
+10
8
1, 3, 18, 80, 75, 126, 196, 144, 405
COMMENTS
We apply every symbol permutation on the representatives of isotopic classes to generate Latin squares of order n and calculate the determinants.
These results are based upon work supported by the National Science Foundation under the grants numbered DMS-1852378 and DMS-1560019.
EXAMPLE
For n=2, the only Latin squares of order 2 are [[1, 2], [2, 1]] and [[2, 1], [1, 2]]. Therefore, the minimum absolute value of the determinants of order 2 Latin squares is 3.
PROG
(Sage)
# Takes a string and turns it into a square matrix of order n
def make_matrix(string, n):
m = []
row = []
for i in range(0, n * n):
if string[i] == '\n':
continue
if string[i] == ' ':
continue
row.append(Integer(string[i]) + 1)
if len(row) == n:
m.append(row)
row = []
return matrix(m)
# Reads a file and returns a list of the matrices in the file
def fetch_matrices(file_name, n):
matrices = []
with open(file_name) as f:
L = f.readlines()
for i in L:
matrices.append(make_matrix(i, n))
return matrices
# Takes a matrix and permutates each symbol in the matrix
# with the given permutation
def permute_matrix(matrix, permutation, n):
copy = deepcopy(matrix)
for i in range(0, n):
for j in range(0 , n):
copy[i, j] = permutation[copy[i][j] - 1]
return copy
"""
Creates a determinant list with the following triples,
[Isotopy Class Representative, Permutation, Determinant]
The Isotopy class representatives come from a file that
contains all Isotopy classes.
"""
def create_determinant_list(file_name, n):
the_list = []
permu = (Permutations(n)).list()
matrices = fetch_matrices(file_name, n)
for i in range(0, len(matrices)):
for j in permu:
copy = permute_matrix(matrices[i], j, n)
the_list.append([i, j, copy.determinant()])
print(len(the_list))
return the_list
Maximum determinant of an n X n Latin square.
+10
8
1, 1, 3, 18, 160, 2325, 41895, 961772, 26978400, 929587995
COMMENTS
a(10) = 36843728625, conjectured. See Stack Exchange link. - Hugo Pfoertner, Sep 29 2019
It is unknown, but very likely, that A301371(n) > a(n) also holds for all n > 9 - Hugo Pfoertner, Dec 12 2020
EXAMPLE
An example of an 8 X 8 Latin square with maximum determinant is
[7 1 3 4 8 2 5 6]
[1 7 4 3 6 5 2 8]
[3 4 1 7 2 6 8 5]
[4 3 7 1 5 8 6 2]
[8 6 2 5 4 7 1 3]
[2 5 6 8 7 3 4 1]
[5 2 8 6 1 4 3 7]
[6 8 5 2 3 1 7 4].
An example of a 9 X 9 Latin square with maximum determinant is
[9 4 3 8 1 5 2 6 7]
[3 9 8 5 4 6 1 7 2]
[4 1 9 3 2 8 7 5 6]
[1 2 4 9 7 3 6 8 5]
[8 3 5 6 9 7 4 2 1]
[2 7 1 4 6 9 5 3 8]
[5 8 6 7 3 2 9 1 4]
[7 6 2 1 5 4 8 9 3]
[6 5 7 2 8 1 3 4 9].
An example of a 10 X 10 Latin square with abs(determinant) = 36843728625 is a circulant matrix with first row [1, 3, 7, 9, 8, 6, 5, 4, 2, 10], but it is not known if this is the best possible. - Kebbaj Mohamed Reda, Nov 27 2019 (reworded by Hugo Pfoertner)
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