A000170 -id:A000170 - cp's OEIS frontend

A140393 Erroneous duplicate of A000170, see comments.

1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 365596, 2279184, 14772512, 95815104, 666090624, 4968057848, 39029188884, 314666222712, 2691008701644, 24233937684440, 226732487925864

Offset: 1

N. J. A. Sloane, Jun 18 2008

dead

This is a new calculation of A000170 which produced a different answer for n=24. I do not know which version is correct and I have therefore created this second entry for the values found by the NQueens(AT)home team. See the main entry for this sequence, A000170, for many further references and comments.

As discovered on Nov 07 2008, this sequence is likely to suffer from an overflow error in the implementation of the NQueens(AT)Home project. The results differ by 781684047872 = 0xb600000000 = 182 * 2^32.

NQueens(AT)home, Home Page [Via Internet Archive Wayback-Machine]

See A000170 for further information.

a(24) is probably mistaken. - Thomas B. Preußer, Nov 07 2008

A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442, 129140163, 258280326, 387420489

Offset: 0

Henry Bottomley, May 03 2000

In general, for the recurrence a(n) = a(n-1)*a(n-2)/a(n-3), all terms are integers iff a(0) divides a(2) and first three terms are positive integers, since a(2n+k) = a(k)*(a(2)/a(0))^n for all nonnegative integers n and k.
Equals eigensequence of triangle A070909; (1, 1, 2, 3, 6, 9, 18, ...) shifts to the left with multiplication by triangle A070909. - Gary W. Adamson, May 15 2010
The a(n) represent all paths of length (n+1), n >= 0, starting at the initial node on the path graph P_5, see the second Maple program. - Johannes W. Meijer, May 29 2010
a(n) is the difference between numbers of multiple of 3 evil (A001969) and odious (A000069) numbers in interval [0, 2^(n+1)). - Vladimir Shevelev, May 16 2012
A "half-geometric progression": to obtain a term (beginning with the third one) we multiply the before previous one by 3. - Vladimir Shevelev, May 21 2012
Pisano periods: 1, 2, 1, 4, 8, 2, 12, 4, 1, 8, 10, 4, 6, 12, 8, 8, 32, 2, 36, 8, ... . - R. J. Mathar, Aug 10 2012
Numbers k such that the k-th cyclotomic polynomial has a root mod 3. - Eric M. Schmidt, Jul 31 2013
Range of row n of the circular Pascal array of order 6. - Shaun V. Ault, Jun 05 2014
Also, the number of walks of length n on the graph 0--1--2--3--4 starting at vertex 1. - Sean A. Irvine, Jun 03 2025

			In the interval [0,2^5) we have 11 multiples of 3 numbers, from which 10 are evil and only one (21) is odious. Thus a(4) = 10 - 1 = 9. - _Vladimir Shevelev_, May 16 2012

Indranil Ghosh, Table of n, a(n) for n = 0..1500 (first 401 terms from T. D. Noe)
S. V. Ault and C. Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Vol. 332 (2014), pp. 45-54.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Sean A. Irvine, Walks on Graphs.
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See pages 106-7.
Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000045, A000069, A000079, A000120, A000244, A001969, A006720, A006721.
Cf. A006722, A006723, A008776, A028495, A030436, A037124, A048328, A061551.
Cf. A068911, A070909, A087503, A124302, A128588, A133464, A140730, A140740.
Cf. A178381, A182751, A182752, A182753, A182754, A182755, A182756, A182757.
Fifth row of the array A094718.

import Data.List (transpose)
a038754 n = a038754_list !! n
a038754_list = concat $ transpose [a000244_list, a008776_list]
-- Reinhard Zumkeller, Oct 19 2015

[n le 2 select n else 3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 18 2016

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-2]+2 od: seq(a[n]+1, n=0..34); # Zerinvary Lajos, Mar 20 2008
with(GraphTheory): P:=5: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=35; for n from 1 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..P) od: seq(a(n),n=1..nmax); # Johannes W. Meijer, May 29 2010

LinearRecurrence[{0,3},{1,2},40] (* Harvey P. Dale, Jan 26 2014 *)
CoefficientList[Series[(1+2x)/(1-3x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2016 *)
Module[{nn=20,c},c=3^Range[0,nn];Riffle[c,2c]] (* Harvey P. Dale, Aug 21 2021 *)

PARI
```
a(n)=(1/6)*(5-(-1)^n)*3^floor(n/2)
```
PARI
```
a(n)=3^(n>>1)<
				
```

[2^(n%2)*3^((n-(n%2))/2) for n in range(61)] # G. C. Greubel, Oct 10 2022

a(n) = a(n-1)*a(n-2)/a(n-3) with a(0)=1, a(1)=2, a(2)=3.
a(2*n) = (3/2)*a(2*n-1) = 3^n, a(2*n+1) = 2*a(2*n) = 2*3^n.
From Benoit Cloitre, Apr 27 2003: (Start)
a(1)=1, a(n)= 2*a(n-1) if a(n-1) is odd, or a(n)= (3/2)*a(n-1) if a(n-1) is even.
a(n) = (1/6)*(5-(-1)^n)*3^floor(n/2).
a(2*n) = a(2*n-1) + a(2*n-2) + a(2*n-3).
a(2*n+1) = a(2*n) + a(2*n-1). (End)
G.f.: (1+2*x)/(1-3*x^2). - Paul Barry, Aug 25 2003
From Reinhard Zumkeller, Sep 11 2003: (Start)
a(n) = (1 + n mod 2) * 3^floor(n/2).
a(n) = A087503(n) - A087503(n-1). (End)
a(n) = sqrt(3)*(2+sqrt(3))*(sqrt(3))^n/6 - sqrt(3)*(2-sqrt(3))*(-sqrt(3))^n/6. - Paul Barry, Sep 16 2003
From Reinhard Zumkeller, May 26 2008: (Start)
a(n) = A140740(n+2,2).
a(n+1) = a(n) + a(n - n mod 2). (End)
If p(i) = Fibonacci(i-3) and if A is the Hessenberg matrix of order n defined by A(i,j) = p(j-i+1), (i<=j), A(i,j)=-1, (i=j+1), and A(i,j)=0 otherwise. Then, for n>=1, a(n-1) = (-1)^n det A. - Milan Janjic, May 08 2010
a(n) = A182751(n) for n >= 2. - Jaroslav Krizek, Nov 27 2010
a(n) = Sum_{i=0..2^(n+1), i==0 (mod 3)} (-1)^A000120(i). - Vladimir Shevelev, May 16 2012
a(0)=1, a(1)=2, for n>=3, a(n)=3*a(n-2). - Vladimir Shevelev, May 21 2012
Sum_(n>=0) 1/a(n) = 9/4. - Alexander R. Povolotsky, Aug 24 2012
a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)). - Richard R. Forberg, Sep 04 2013
a(n) = 2^((1-(-1)^n)/2)*3^((2*n-1+(-1)^n)/4). - Luce ETIENNE, Aug 11 2014
From Reinhard Zumkeller, Oct 19 2015: (Start)
a(2*n) = A000244(n), a(2*n+1) = A008776(n).
For n > 0: a(n+1) = a(n) + if a(n) odd then min{a(n), a(n-1)} else max{a(n), a(n-1)}, see also A128588. (End)
E.g.f.: (7*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x) - 4)/3. - Stefano Spezia, Feb 17 2022
Sum_{n>=0} (-1)^n/a(n) = 3/4. - Amiram Eldar, Dec 02 2022

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

A051223 Number of ways of placing n nonattacking superqueens on an n X n board.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 44, 156, 1876, 5180, 32516, 202900, 1330622, 8924976, 64492432, 495864256, 3977841852, 34092182276, 306819842212, 2883202816808, 28144109776812, 286022102245804

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

A superqueen moves like a queen and a knight.
A linear-time algorithm giving an explicit solution for any n >= 10 for the n-super-queens-problem can be found at the link. Included is an online solver, implemented in JavaScript. - Frank Schwellinger (nummer_eins(AT)web.de), Mar 19 2004. [But see the next comment - N. J. A. Sloane, Jul 01 2021]
Escamocher and O'Sullivan (2021) claim Schwellinger's algorithm is incorrect, and that their own algorithm is the first published linear-time algorithm. - N. J. A. Sloane, Jul 01 2021

D. Bill, Durango Bill's The N-Queens Problem
Guillaume Escamocher and Barry O'Sullivan, Leprechauns on the chessboard, Discrete Mathematics, 344(5), 2021. See especially page 3.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed.
R. Oprisch, N x N SuperQueens Solutions Table
F. Schwellinger, Explicit Solution To the N-Super-Queens Problem.
W. Schubert, N-Queens page

Cf. A051224, A000170.

a(20) from Bill link added Jul 25 2006
a(21)-a(23) from R. Oprisch's website added by Max Alekseyev, Sep 29 2006
a(24)-a(26) from W. Schubert, Jul 31 2009, Nov 29 2009, Jan 18 2011

A002562 Number of ways of placing n nonattacking queens on n X n board (symmetric solutions count only once).

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 6, 12, 46, 92, 341, 1787, 9233, 45752, 285053, 1846955, 11977939, 83263591, 621012754, 4878666808, 39333324973, 336376244042, 3029242658210, 28439272956934, 275986683743434, 2789712466510289, 29363495934315694

Offset: 1

N. J. A. Sloane

			a(4) = 1:
  +---------+
  | . . Q . |
  | Q . . . |
  | . . . Q |
  | . Q . . |
  +---------+
a(5) = 2:
  +-----------+ +-----------+
  | . . . Q . | | . . . Q . |
  | . Q . . . | | Q . . . . |
  | . . . . Q | | . . Q . . |
  | . . Q . . | | . . . . Q |
  | Q . . . . | | . Q . . . |
  +-----------+ +-----------+
a(6) = 1:
  +-------------+
  | . . . . Q . |
  | . . Q . . . |
  | Q . . . . . |
  | . . . . . Q |
  | . . . Q . . |
  | . Q . . . . | - _Hugo Pfoertner_, Mar 17 2019
  +-------------+
a(7) = 6:
  +---------------+  +---------------+  +---------------+
  | Q . . . . . . |  | Q . . . . . . |  | . Q . . . . . |
  | . . Q . . . . |  | . . . Q . . . |  | . . . Q . . . |
  | . . . . Q . . |  | . . . . . . Q |  | Q . . . . . . |
  | . . . . . . Q |  | . . Q . . . . |  | . . . . . . Q |
  | . Q . . . . . |  | . . . . . Q . |  | . . . . Q . . |
  | . . . Q . . . |  | . Q . . . . . |  | . . Q . . . . |
  | . . . . . Q . |  | . . . . Q . . |  | . . . . . Q . |
  +---------------+  +---------------+  +---------------+
.
  +---------------+  +---------------+  +---------------+
  | . Q . . . . . |  | . Q . . . . . |  | . Q . . . . . |
  | . . . . Q . . |  | . . . . Q . . |  | . . . . . Q . |
  | Q . . . . . . |  | . . . . . . Q |  | . . Q . . . . |
  | . . . Q . . . |  | . . . Q . . . |  | . . . . . . Q |
  | . . . . . . Q |  | Q . . . . . . |  | . . . Q . . . |
  | . . Q . . . . |  | . . Q . . . . |  | Q . . . . . . |
  | . . . . . Q . |  | . . . . . Q . |  | . . . . Q . . |
  +---------------+  +---------------+  +---------------+
- _Hugo Pfoertner_, Mar 18 2019

Martin Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, p. 231-233.
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. 238.

J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656. [Annotated scanned copy]
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656.
P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
V. Chvatal, All solutions to the problem of eight queens
V. Chvatal, All solutions to the problem of eight queens [Cached copy, pdf format, with permission]
Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-1. Illustrates a(8)=12.
Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-2.
Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-3.
Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-4.
Thomas Preusser, Queens%40TUD-Project
E. M. Reingold, Letter to N. J. A. Sloane, Dec 27 1973
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47. [Incomplete annotated scan of title page and pages 18-51]
Eric Weisstein's World of Mathematics, Queens Problem.
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Wikipedia, Eight queens puzzle.

Cf. A000170, A032522, A033148.

a(n) = (1/8) * (Q(n) + P(n) + 2 * R(n)), where Q(n) = A000170(n) [all solutions], P(n) = A032522(n) [point symmetric solutions] and R(n) = A033148(n) [rotationally symmetric solutions].

a(17) and a(18) found by Ulrich Schimke in Goettingen, Germany (UlrSchimke(AT)aol.com)
Formula and a(19) to a(23) added by Matthias Engelhardt in Nuremberg, Germany, Jan 23 2000
Terms (calculated from formula) added by Thomas B. Preußer, Dec 15 2008
a(26) (derived from formula after recent extension of A000170) added by Thomas B. Preußer, Jul 12 2009
a(27) (derived from formula after recent extension of A000170) added by Thomas B. Preußer, Sep 23 2016

A000136 Number of ways of folding a strip of n labeled stamps.

Original entry on oeis.org

1, 2, 6, 16, 50, 144, 462, 1392, 4536, 14060, 46310, 146376, 485914, 1557892, 5202690, 16861984, 56579196, 184940388, 622945970, 2050228360, 6927964218, 22930109884, 77692142980, 258360586368, 877395996200, 2929432171328, 9968202968958, 33396290888520, 113837957337750

Offset: 1

N. J. A. Sloane

nonn

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. 238.

T. D. Noe, Table of n, a(n) for n = 1..45
Oswin Aichholzer, Florian Lehner, and Christian Lindorfer, Folding polyominoes into cubes, arXiv:2402.14965 [cs.CG], 2024. See p. 9.
T. Asano, E. D. Demaine, M. L. Demaine and R. Uehara, NP-completeness of generalized Kaboozle, J. Information Processing, 20 (July, 2012), 713-718.
CombOS - Combinatorial Object Server, Generate meanders and stamp foldings
R. Dickau, Stamp Folding
R. Dickau, Stamp Folding [Cached copy, pdf format, with permission]
Thomas C. Hull, Adham Ibrahim, Jacob Paltrowitz, Natalya Ter-Saakov, and Grace Wang, The Stamp Folding Problem From a Mountain-Valley Perspective: Enumerations and Bounds, arXiv:2503.23661 [math.CO], 2025. See p. 1.
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152.
J. E. Koehler, Folding a strip of stamps, J. Combin. Theory, 5 (1968), 135-152. [Annotated, corrected, scanned copy]
Stéphane Legendre, The 16 foldings of 4 labeled stamps
Bowie Liu, Dennis Wong, Chan-Tong Lam, and Marcus Im, Recursive and iterative approaches to generate rotation Gray codes for stamp foldings and semi-meanders, arXiv:2411.05458 [cs.DS], 2024. See p. 2.
W. F. Lunnon, A map-folding problem, Math. Comp. 22 (1968), 193-199.
David Orden, In how many ways can you fold a strip of stamps?, 2014.
A. Panayotopoulos and P. Vlamos, Partitioning the Meandering Curves, Mathematics in Computer Science (2015) p 1-10.
Frank Ruskey, Information on Stamp Foldings
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 41.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 41. [Incomplete annotated scan of title page and pages 18-51]
J. Sawada and R. Li, Stamp foldings, semi-meanders, and open meanders: fast generation algorithms, Electronic Journal of Combinatorics, Volume 19 No. 2 (2012), P#43 (16 pages).
Eric Weisstein's World of Mathematics, Stamp Folding
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Index entries for sequences obtained by enumerating foldings

Cf. A000560, A000682.

a(n) = 2*n * A000560(n-1) for n >= 3.

a(n) = n * A000682(n). - Andrew Howroyd, Dec 06 2015

A274650 Triangle read by rows: T(n,k), (0 <= k <= n), in which each term is the least nonnegative integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 2, 4, 3, 5, 5, 1, 0, 2, 3, 4, 3, 5, 1, 6, 7, 6, 7, 2, 0, 5, 4, 8, 8, 5, 9, 4, 7, 2, 10, 6, 7, 10, 8, 3, 0, 6, 9, 5, 4, 11, 6, 12, 7, 1, 8, 3, 10, 9, 13, 9, 8, 4, 11, 2, 0, 1, 12, 6, 7, 10, 10, 11, 7, 12, 4, 3, 2, 9, 8, 14, 13, 15, 12, 9, 10, 6, 8, 1, 0, 11, 7, 4, 16, 14, 17

Offset: 0

Omar E. Pol, Jul 02 2016

Similar to A274528, but the triangle here is a right triangle.
The same rule applied to an equilateral triangle gives A274528.
By analogy, the offset for both rows and columns is the same as the offset of A274528.
We construct the triangle by reading from left to right in each row, starting with T(0,0) = 0.
Presumably every diagonal and every column is also a permutation of the nonnegative integers, but the proof does not seem so straightforward. Of course neither the rows nor the antidiagonals are permutations of the nonnegative integers, since they are finite in length.
Omar E. Pol's conjecture that every column and every diagonal of the triangle is a permutation of the nonnegative integers is true: see the link. - N. J. A. Sloane, Jun 07 2017
It appears that the numbers generally appear for the first time in or near the right border of the triangle.
Theorem 1: the middle diagonal gives A000004 (the zero sequence).
Proof: T(0,0) = 0 by definition. For the next rows we have that in row 1 there are no zeros because the first term belongs to a column that contains a zero and the second term belongs to a diagonal that contains a zero. In row 2 the unique zero is T(2,1) = 0 because the preceding term belongs to a column that contains a zero and the following term belongs to a diagonal that contains a zero. Then we have two recurrences for all rows of the triangle:
a) If T(n,k) = 0 then row n+1 does not contain a zero because every term belongs to a column that contains a zero or it belongs to a diagonal that contains a zero.
b) If T(n,k) = 0 the next zero is T(n+2,k+1) because every preceding term in row n+2 is a positive integer because it belongs to a column that contains a zero and, on the other hand, the column, the diagonal and the antidiagonal of T(n+2,k+1) do not contain zeros.
Finally, since both T(n,k) = 0 and T(n+2,k+1) = 0 are located in the middle diagonal, the other terms of the middle diagonal are zeros, or in other words: the middle diagonal gives A000004 (the zero sequence). QED
Theorem 2: all zeros are in the middle diagonal.
Proof: consider the first n rows of the triangle. Every element located above or at the right-hand side of the middle diagonal must be positive because it belongs to a diagonal that contains one of the zeros of the middle diagonal. On the other hand every element located below the middle diagonal must be positive because it belongs to a column that contains one of the zeros of the middle diagonal, hence there are no zeros outside of the middle diagonal, or in other words: all zeros are in the middle diagonal. QED
From Hartmut F. W. Hoft, Jun 12 2017: (Start)
T(2k,k) = 0, for all k >= 0, and T(n,{(n-1)/2,(n+2)/2,(n-2)/2,(n+1)/2}) = 1, for all n >= 0 with n mod 8 = {1,2,4,5} respectively, and no 0's or 1's occur in other positions. The triangle of positions of 0's and 1's for this sequence is the triangle in the Comment section of A274651 with row and column indices and values shifted down by one.
The sequence of rows containing 1's is A047613 (n mod 8 = {1,2,4,5}), those containing only 1's is A016813 (n mod 8 = {1,5}), those containing both 0's and 1's is A047463 (n mod 8 = {2,4}), those containing only 0's is A047451 (n mod 8 = {0,6}), and those containing neither 0's nor 2's is A004767 (n mod 8 = {3,7}).
(End)

			Triangle begins:
   0;
   1,  2;
   3,  0,  1;
   2,  4,  3,  5;
   5,  1,  0,  2,  3;
   4,  3,  5,  1,  6,  7;
   6,  7,  2,  0,  5,  4,  8;
   8,  5,  9,  4,  7,  2, 10,  6;
   7, 10,  8,  3,  0,  6,  9,  5,  4;
  11,  6, 12,  7,  1,  8,  3, 10,  9, 13;
   9,  8,  4, 11,  2,  0,  1, 12,  6,  7, 10;
  10, 11,  7, 12,  4,  3,  2,  9,  8, 14, 13, 15;
  12,  9, 10,  6,  8,  1,  0, 11,  7,  4, 16, 14, 17;
  ...
From _Omar E. Pol_, Jun 07 2017: (Start)
The triangle may be reformatted as an isosceles triangle so that the zero sequence (A000004) appears in the central column (but note that this is NOT the way the triangle is constructed!):
.
.                    0;
.                  1,  2;
.                3,  0,  1;
.              2,  4,  3,  5;
.            5,  1,  0,  2,  3;
.          4,  3,  5,  1,  6,  7;
.        6,  7,  2,  0,  5,  4,  8;
.     8,   5,  9,  4,  7,  2, 10,  6;
.   7,  10,  8,  3,  0,  6,  9,  5,  4;
...
(End)

Alois P. Heinz, Rows n = 0..200, flattened
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Rémy Sigrist, Colored representation of the rows n = 0..999
Rémy Sigrist, PARI program for A274650
N. J. A. Sloane, Notes on A274650 and Proof by Non-Attacking Queens

Cf. A000004 (middle diagonal).
Cf. A046092 (indices of the zeros).
Every diagonal and every column of the right triangle is a permutation of A001477.
The left and right edges of the right triangle give A286294 and A286295.
Cf. A274651 is the same triangle but with 1 added to every entry.
Other sequences of the same family are A269526, A274528, A274820, A274821, A286297, A288530, A288531.
Sequences mentioned in N. J. A. Sloane's proof are A000170, A274616 and A287864.
Cf. A288384.
Cf. A004767, A016813, A047451, A047463, A047613. - Hartmut F. W. Hoft, Jun 12 2017
See A308179, A308180 for a very similar triangle.

(* function a274651[] is defined in A274651 *)
(* computation of rows 0 ... n-1 *)
a274650[n_] := a274651[n]-1
Flatten[a274650[13]] (* data *)
TableForm[a274650[13]] (* triangle *)
(* Hartmut F. W. Hoft, Jun 12 2017 *)

PARI
```
See Links section.
```

T(n,k) = A274651(n+1,k+1) - 1.

A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 4, 6, 0, 2, 16, 20, 24, 0, 14, 44, 80, 108, 120, 0, 90, 200, 384, 544, 672, 720, 0, 646, 1288, 2240, 3264, 4128, 4800, 5040, 0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320, 0, 47622, 78652, 123456, 176832, 231936, 280512, 323520, 352800, 362880

Offset: 0

Alois P. Heinz, Apr 02 2020

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = n! for k>=n.

			Triangle T(n,k) begins:
  1;
  0,    1;
  0,    0,    2;
  0,    0,    4,     6;
  0,    2,   16,    20,    24;
  0,   14,   44,    80,   108,   120;
  0,   90,  200,   384,   544,   672,   720;
  0,  646, 1288,  2240,  3264,  4128,  4800,  5040;
  0, 5242, 9512, 15424, 23040, 28992, 34752, 38880, 40320;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11.
Wikipedia, Permutation

Columns k=0-10 (for n>=k) give: A000007, A002464, A110128, A117574, A189255, A189256, A189271, A360384, A360386, A360462, A360463.
Main diagonal gives A000142.
T(2n,n) gives A189849.
T(n+1,n) gives 4*A138772(n).
T(n+2,n) gives 16*A333804(n).
Cf. A000170 (condition is satisfied for all k), A248686 (p(i) at distance k are sorted).

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

A137774 Number of ways to place n nonattacking empresses on an n X n board.

Original entry on oeis.org

1, 2, 2, 8, 20, 94, 438, 2766, 19480, 163058, 1546726, 16598282, 197708058, 2586423174, 36769177348, 563504645310, 9248221393974, 161670971937362, 2996936692836754, 58689061747521430, 1210222434323163704, 26204614054454840842, 594313769819021397534, 14086979362268860896282

Offset: 1

Vaclav Kotesovec, Jan 27 2011

An empress moves like a rook and a knight.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Separators - a new statistic for permutations, arXiv:1905.12364 [math.CO], 2019.
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, On the Sparseness of the Downsets of Permutations via Their Number of Separators, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 3, Article #S2R21.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.685 and 636.
W. Schubert, N-Queens page

Cf. A201513, A000170, A002465, A201540.
Cf. A185085, A051223, A244284, A201511, A201861, A137774, A245011.
Cf. A218244, A002464, A110128, A117574, A089222, A002493.

Asymptotics (Vaclav Kotesovec, Jan 26 2011): a(n)/n! -> 1/e^4.
General asymptotic formulas for number of ways to place n nonattacking pieces rook + leaper[r,s] on an n X n board:
a(n)/n! -> 1/e^2 for 0a(n)/n! -> 1/e^4 for 0

Terms a(16)-a(17) from Vaclav Kotesovec, Feb 06 2011
Terms a(18)-a(19) from Wolfram Schubert, Jul 24 2011
Terms a(20)-a(24) (computed by Wolfram Schubert), Vaclav Kotesovec, Aug 25 2012

Showing 1-10 of 89 results. Next

cp's OEIS Frontend

A140393 Erroneous duplicate of A000170, see comments.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A051223 Number of ways of placing n nonattacking superqueens on an n X n board.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A002562 Number of ways of placing n nonattacking queens on n X n board (symmetric solutions count only once).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Formula

Extensions

A000136 Number of ways of folding a strip of n labeled stamps.

Views

Author

Keywords

References

Links

Crossrefs

Formula

A274650 Triangle read by rows: T(n,k), (0 <= k <= n), in which each term is the least nonnegative integer such that no row, column, diagonal, or antidiagonal contains a repeated term.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A333706 Number T(n,k) of permutations p of [n] such that |p(i+k) - p(i)| <> k for i in [n-k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples