A003682 -id:A003682 - cp's OEIS frontend

A014206 a(n) = n^2 + n + 2.

2, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158, 184, 212, 242, 274, 308, 344, 382, 422, 464, 508, 554, 602, 652, 704, 758, 814, 872, 932, 994, 1058, 1124, 1192, 1262, 1334, 1408, 1484, 1562, 1642, 1724, 1808, 1894, 1982, 2072, 2164, 2258, 2354, 2452, 2552

Offset: 0

N. J. A. Sloane

Draw n + 1 circles in the plane; sequence gives maximal number of regions into which the plane is divided. Cf. A051890, A386480.
Number of binary (zero-one) bitonic sequences of length n + 1. - Johan Gade (jgade(AT)diku.dk), Oct 15 2003
Also the number of permutations of n + 1 which avoid the patterns 213, 312, 13452 and 34521. Example: the permutations of 4 which avoid 213, 312 (and implicitly 13452 and 34521) are 1234, 1243, 1342, 1432, 2341, 2431, 3421, 4321. - Mike Zabrocki, Jul 09 2007
If Y is a 2-subset of an n-set X then, for n >= 3, a(n-3) is equal to the number of (n-3)-subsets and (n-1)-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
With a different offset, competition number of the complete tripartite graph K_{n, n, n}. [Kim, Sano] - Jonathan Vos Post, May 14 2009. Cf. A160450, A160457.
A related sequence is A241119. - Avi Friedlich, Apr 28 2015
From Avi Friedlich, Apr 28 2015: (Start)
This sequence, which also represents the number of Hamiltonian paths in K_2 X P_n (A200182), may be represented by interlacing recursive polynomials in arithmetic progression (discriminant =-63). For example:
a(3*k-3) = 9*k^2 - 15*k + 8,
a(3*k-2) = 9*k^2 - 9*k  + 4,
a(3*k-1) = 9*k^2 - 3*k  + 2,
a(3*k)   = 3*(k+1)^2  - 1. (End)
a(n+1) is the area of a triangle with vertices at (n+3, n+4), ((n-1)*n/2, n*(n+1)/2),((n+1)^2, (n+2)^2) with n >= -1. - J. M. Bergot, Feb 02 2018
For prime p and any integer k, k^a(p-1) == k^2 (mod p^2). - Jianing Song, Apr 20 2019
From Bernard Schott, Jan 01 2021: (Start)
For n >= 1, a(n-1) is the number of solutions x in the interval 0 <= x <= n of the equation x^2 - [x^2] = (x - [x])^2, where [x] = floor(x). For n = 3, the a(2) = 8 solutions in the interval [0, 3] are 0, 1, 3/2, 2, 9/4, 5/2, 11/4 and 3.
This is a variant of the 4th problem proposed during the 20th British Mathematical Olympiad in 1984 (see A002061). The interval [1, n] of the Olympiad problem becomes here [0, n], and only the new solution x = 0 is added. (End)
See A386480 for the almost identical sequence 1, 2, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, ... which is the maximum number of regions that can be formed in the plane by drawing n circles, and the maximum number of regions that can be formed on the sphere by drawing n great circles. - N. J. A. Sloane, Aug 01 2025

			a(0) = 0^2 + 0 + 2 = 2.
a(1) = 1^2 + 1 + 2 = 4.
a(2) = 2^2 + 2 + 2 = 8.
a(6) = 4*5/5 + 5*6/5 + 6*7/5 + 7*8/5 + 8*9/5 = 44. - _Bruno Berselli_, Oct 20 2016

K. E. Batcher, Sorting Networks and their Applications. Proc. AFIPS Spring Joint Comput. Conf., Vol. 32, pp. 307-314 (1968). [for bitonic sequences]
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 3.
T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms. MIT Press / McGraw-Hill (1990) [for bitonic sequences]
Indiana School Mathematics Journal, vol. 14, no. 4, 1979, p. 4.
D. E. Knuth, The Art of Computer Programming, vol3: Sorting and Searching, Addison-Wesley (1973) [for bitonic sequences]
J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 177.
Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #44 (First published: San Francisco: Holden-Day, Inc., 1964)

N. J. A. Sloane, Table of n, a(n) for n = 0..1000
A. Burstein, S. Kitaev, and T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
S.-R. Kim and Y. Sano, The competition numbers of complete tripartite graphs, Discrete Appl. Math., 156 (2008) 3522-3524.
Hans Werner Lang, Bitonic sequences.
Daniel Q. Naiman and Edward R. Scheinerman, Arbitrage and Geometry, arXiv:1709.07446 [q-fin.MF], 2017.
Jean-Christoph Novelli and Anne Schilling, The Forgotten Monoid, arXiv 0706.2996 [math.CO], 2007.
Parabola, Problem #Q736, 24(1) (1988), p. 22.
Franck Ramaharo, Enumerating the states of the twist knot, arXiv:1712.06543 [math.CO], 2017.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Yoshio Sano, The competition numbers of regular polyhedra, arXiv:0905.1763 [math.CO], 2009.
Jeffrey Shallit, Recursivity: An Interesting but Little-Known Function, 2012. [Mentions this function in a blog post as the solution for small n to a problem involving Boolean matrices whose values for larger n are unknown.]
Eric Weisstein's World of Mathematics, Plane Division by Circles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).
A row of A059250.
Cf. A000124, A051890, A002522, A241119, A033547 (partial sums).
Cf. A002061 (central polygonal numbers).
Cf. A003682, A160450, A160457, A200182, A386480.
Column 4 of A347570.

[n^2+n+2: n in [0..50]]; // Vincenzo Librandi, Apr 29 2015

Maple
```
A014206 := n->n^2+n+2;
```

Table[n^2 + n + 2, {n, 0, 50}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{3, -3, 1}, {2, 4, 8}, 50] (* Harvey P. Dale, May 14 2011 *)
CoefficientList[Series[2 (x^2 - x + 1)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 29 2015 *)

a(n)=n^2+n+2 \\ Charles R Greathouse IV, Jul 31 2011

x='x+O('x^100); Vec(2*x*(x^2-x+1)/(1-x)^3) \\ Altug Alkan, Nov 01 2015

G.f.: 2*(x^2 - x + 1)/(1 - x)^3.
n hyperspheres divide R^k into at most C(n-1, k) + Sum_{i = 0..k} C(n, i) regions.
a(n) = A002061(n+1) + 1 for n >= 0. - Rick L. Shepherd, May 30 2005
Equals binomial transform of [2, 2, 2, 0, 0, 0, ...]. - Gary W. Adamson, Jun 18 2008
a(n) = A003682(n+1), n > 0. - R. J. Mathar, Oct 28 2008
a(n) = a(n-1) + 2*n (with a(0) = 2). - Vincenzo Librandi, Nov 20 2010
a(0) = 2, a(1) = 4, a(2) = 8, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. - Harvey P. Dale, May 14 2011
a(n + 1) = n^2 + 3*n + 4. - Alonso del Arte, Apr 12 2015
a(n) = Sum_{i=n-2..n+2} i*(i + 1)/5. - Bruno Berselli, Oct 20 2016
Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(7)/2)/sqrt(7). - Amiram Eldar, Jan 09 2021
From Amiram Eldar, Jan 29 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(11)*Pi/2)*sech(sqrt(7)*Pi/2).
Product_{n>=0} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)*sech(sqrt(7)*Pi/2). (End)
a(n) = 2*A000124(n). - R. J. Mathar, Mar 14 2021
E.g.f.: exp(x)*(2 + 2*x + x^2). - Stefano Spezia, Apr 30 2022

More terms from Stefan Steinerberger, Apr 08 2006

A332307 Array read by antidiagonals: T(m,n) is the number of (undirected) Hamiltonian paths in the m X n grid graph.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 14, 20, 14, 1, 1, 22, 62, 62, 22, 1, 1, 32, 132, 276, 132, 32, 1, 1, 44, 336, 1006, 1006, 336, 44, 1, 1, 58, 688, 3610, 4324, 3610, 688, 58, 1, 1, 74, 1578, 12010, 26996, 26996, 12010, 1578, 74, 1, 1, 92, 3190, 38984, 109722, 229348, 109722, 38984, 3190, 92, 1

Offset: 1

Andrew Howroyd, Feb 09 2020

			Array begins:
================================================
m\n | 1  2   3     4      5       6        7
----+-------------------------------------------
  1 | 1  1   1     1      1       1        1 ...
  2 | 1  4   8    14     22      32       44 ...
  3 | 1  8  20    62    132     336      688 ...
  4 | 1 14  62   276   1006    3610    12010 ...
  5 | 1 22 132  1006   4324   26996   109722 ...
  6 | 1 32 336  3610  26996  229348  1620034 ...
  7 | 1 44 688 12010 109722 1620034 13535280 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..435
J. L. Jacobsen, Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions, J. Phys. A: Math. Theor. 40 (2007) 14667-14678.
J.-M. Mayer, C. Guez and J. Dayantis, Exact computer enumeration of the number of Hamiltonian paths in small square plane lattices, Physical Review B, Vol. 42 Number 1, 1990.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path

Rows n=1..9 are A000012, A003682, A003685, A003695, A003778, A145402, A358794, A358795, A358796.
Main diagonal is A120443.
Cf. A064298, A231829, A271465, A271592, A288518, A321172.

T(n,m) = T(m,n).

A137882 Number of (directed) Hamiltonian paths in the n-ladder graph.

Original entry on oeis.org

2, 8, 16, 28, 44, 64, 88, 116, 148, 184, 224, 268, 316, 368, 424, 484, 548, 616, 688, 764, 844, 928, 1016, 1108, 1204, 1304, 1408, 1516, 1628, 1744, 1864, 1988, 2116, 2248, 2384, 2524, 2668, 2816, 2968, 3124, 3284, 3448, 3616, 3788, 3964, 4144, 4328, 4516, 4708, 4904, 5104, 5308, 5516, 5728, 5944, 6164, 6388, 6616, 6848, 7084, 7324, 7568, 7816

Offset: 1

Eric W. Weisstein, Feb 20 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Ladder Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001221, A001222, A003682, A154685.

A137882:=n->2*(n^2-n+2): 2,seq(A137882(n), n=2..100); # Wesley Ivan Hurt, Apr 25 2017

CoefficientList[Series[2*x*(1 + x - x^2 + x^3)/(1 - x)^3, {x,0,50}], x] (* G. C. Greubel, Apr 25 2017 *)
LinearRecurrence[{3,-3,1},{2,8,16,28},70] (* Harvey P. Dale, Nov 15 2018 *)

my(x='x+O('x^50)); Vec(2*x*(1 + x - x^2 + x^3)/(1 - x)^3) \\ G. C. Greubel, Apr 25 2017

For n>2, m = p^3*q (p,q = primes), a(n) = Sum_{d|m} (n-1)^(bigomega(d) - omega(d)) = Sum_{d|m} (n-1)^(A001222(d) - A001221(d)). - Jaroslav Krizek, Sep 24 2009
For n>1, a(n) = 2*(n^2 - n + 2); first diagonal of A154685. - Vincenzo Librandi, Nov 24 2010
G.f.: 2*x*(1+x-x^2+x^3)/(1-x)^3. - Colin Barker, Jan 20 2012
Sum_{n>=1} 1/a(n) = 1/4 + Pi*tanh(sqrt(7)*Pi/2)/(2*sqrt(7)). - Amiram Eldar, Dec 23 2022
From Elmo R. Oliveira, Jun 06 2025: (Start)
E.g.f.: 2*(exp(x)*(2 + x^2) - (2 + x)).
a(n) = 2*A003682(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. (End)

Extended and formula corrected by Max Alekseyev, Apr 11 2009

Corrected the formula which was confusing offsets - R. J. Mathar, Jun 04 2010

A219686 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, vertical or antidiagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..2 nXk array.

Original entry on oeis.org

3, 3, 3, 6, 4, 6, 10, 8, 9, 10, 15, 14, 31, 19, 15, 21, 22, 87, 99, 35, 21, 28, 32, 208, 427, 269, 60, 28, 36, 44, 452, 1531, 1688, 655, 98, 36, 45, 58, 922, 5031, 8964, 5726, 1506, 154, 45, 55, 74, 1799, 15763, 44736, 44816, 18486, 3356, 234, 55, 66, 92, 3394, 47784

Offset: 1

R. H. Hardin Nov 25 2012

Table starts
..3...3.....6......10........15.........21.........28..........36.........45
..3...4.....8......14........22.........32.........44..........58.........74
..6...9....31......87.......208........452........922........1799.......3394
.10..19....99.....427......1531.......5031......15763.......47784.....140586
.15..35...269....1688......8964......44736.....219216.....1062133....5046678
.21..60...655....5726.....44816.....344926....2709320....21474269..167685059
.28..98..1506...18486....220182....2706117...35011623...461414270.5979059010
.36.154..3356...59458...1116208...22404783..479633094.10451333258
.45.234..7278..189516...5697556..187497531.6609458081
.55.345.15335..589078..28354294.1518906665
.66.495.31362.1771979.135630422
.78.693.62286.5155016

			Some solutions for n=3 k=4
..1..1..1..1....0..0..0..0....1..1..0..0....1..1..0..0....1..1..1..0
..1..1..1..1....0..0..0..0....1..1..0..0....1..0..0..0....1..1..0..0
..2..1..1..1....1..1..0..0....2..2..0..0....2..0..0..0....2..2..0..0

R. H. Hardin, Table of n, a(n) for n = 1..127

Column 1 and row 1 are A000217 for n>1

Row 2 is A003682 for n>1

A221542 T(n,k) = Number of 0..k arrays of length n with each element differing from at least one neighbor by something other than 1, starting with 0.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 4, 2, 0, 4, 8, 10, 3, 0, 5, 14, 30, 22, 5, 0, 6, 22, 68, 103, 54, 8, 0, 7, 32, 130, 303, 364, 134, 13, 0, 8, 44, 222, 716, 1386, 1276, 334, 21, 0, 9, 58, 350, 1455, 4018, 6311, 4483, 822, 34, 0, 10, 74, 520, 2658, 9665, 22466, 28762, 15740, 2014, 55, 0, 11

Offset: 1

R. H. Hardin, Jan 19 2013

Table starts
..0.....0......0........0.........0.........0..........0...........0
..1.....2......3........4.........5.........6..........7...........8
..1.....4......8.......14........22........32.........44..........58
..2....10.....30.......68.......130.......222........350.........520
..3....22....103......303.......716......1455.......2658........4487
..5....54....364.....1386......4018......9665......20386.......39007
..8...134...1276.....6311.....22466.....64047.....156098......338711
.13...334...4483....28762....125701....424593....1195561.....2941622
.21...822..15740...131012....703193...2814515....9156379....25546512
.34..2014..55274...596784...3933916..18656979...70126074...221859676
.55..4934.194095..2718469..22007609.123673887..537074685..1926747595
.89.12110.681576.12383368.123117952.819813575.4113296146.16732904887

			Some solutions for n=6 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..4....2....2....3....2....0....4....0....4....4....3....4....0....4....4....4
..0....4....4....4....0....2....3....2....3....0....0....4....2....1....4....4
..0....0....0....4....4....4....1....3....1....4....2....2....0....1....2....0
..3....2....4....4....4....0....0....3....1....1....4....4....4....2....0....0
..0....2....1....2....0....2....2....3....3....1....4....2....0....0....0....3

R. H. Hardin, Table of n, a(n) for n = 1..2080

Column 1 is A000045(n-1).
Row 2 is A000027.
Row 3 is A003682.
Row 4 is A034262.

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 3*a(n-1) -2*a(n-2) +4*a(n-4)
k=3: a(n) = 3*a(n-1) +2*a(n-2) -a(n-3) +a(n-4)
k=4: a(n) = 5*a(n-1) -3*a(n-2) +a(n-3) +15*a(n-4) +3*a(n-5) for n>6
k=5: a(n) = 5*a(n-1) +3*a(n-2) +9*a(n-4) +6*a(n-5) +3*a(n-6)
k=6: a(n) = 7*a(n-1) -4*a(n-2) +6*a(n-3) +26*a(n-4) +10*a(n-5) +16*a(n-6) +12*a(n-8)
k=7: a(n) = 7*a(n-1) +4*a(n-2) +5*a(n-3) +20*a(n-4) +20*a(n-5) +23*a(n-6) -6*a(n-7) +3*a(n-8)
Empirical for row n:
n=2: a(n) = 1*n for n>1
n=3: a(n) = 1*n^2 - 1*n + 2 for n>1
n=4: a(n) = 1*n^3 + 1*n
n=5: a(n) = 1*n^4 + 1*n^3 - 3*n^2 + 10*n - 9 for n>3
n=6: a(n) = 1*n^5 + 2*n^4 - 6*n^3 + 21*n^2 - 31*n + 23 for n>4
n=7: a(n) = 1*n^6 + 3*n^5 - 8*n^4 + 25*n^3 - 30*n^2 + 20*n - 9 for n>3

A201375 T(n,k)=Number of nXk 0..1 arrays with rows and columns lexicographically nondecreasing read forwards, and nonincreasing read backwards.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 4, 4, 2, 2, 5, 8, 5, 2, 2, 6, 14, 14, 6, 2, 2, 7, 22, 36, 22, 7, 2, 2, 8, 32, 80, 80, 32, 8, 2, 2, 9, 44, 157, 268, 157, 44, 9, 2, 2, 10, 58, 280, 786, 786, 280, 58, 10, 2, 2, 11, 74, 464, 2016, 3739, 2016, 464, 74, 11, 2, 2, 12, 92, 726, 4608, 15574, 15574, 4608

Offset: 1

R. H. Hardin Nov 30 2011

Table starts
.2..2..2....2.....2......2........2.........2...........2............2
.2..3..4....5.....6......7........8.........9..........10...........11
.2..4..8...14....22.....32.......44........58..........74...........92
.2..5.14...36....80....157......280.......464.........726.........1085
.2..6.22...80...268....786.....2016......4608........9582........18446
.2..7.32..157...786...3739....15574.....55410......170616.......465037
.2..8.44..280..2016..15574...115168....728078.....3793342.....16517460
.2..9.58..464..4608..55410...728078...8866257....88486476....701948916
.2.10.74..726..9582.170616..3793342..88486476..1799674094..28852930912
.2.11.92.1085.18446.465037.16517460.701948916.28852930912.976134459840

			Some solutions for n=7 k=5
..0..0..0..1..1....0..0..0..1..1....0..0..1..1..1....0..0..0..1..1
..0..0..0..1..1....0..0..0..1..1....0..1..0..1..1....0..0..0..1..1
..0..0..0..1..1....1..1..1..0..1....0..1..1..0..1....0..1..1..0..1
..0..0..1..0..1....1..1..1..0..1....0..1..1..0..1....1..0..1..0..1
..0..1..0..0..1....1..1..1..1..0....0..1..1..1..0....1..0..1..1..0
..1..0..0..0..1....1..1..1..1..0....0..1..1..1..0....1..1..0..0..0
..1..1..1..1..0....1..1..1..1..0....1..0..0..0..0....1..1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 3 is A003682

A386480 a(0) = 1; thereafter a(n) = n^2 - n + 2.

Original entry on oeis.org

1, 2, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158, 184, 212, 242, 274, 308, 344, 382, 422, 464, 508, 554, 602, 652, 704, 758, 814, 872, 932, 994, 1058, 1124, 1192, 1262, 1334, 1408, 1484, 1562, 1642, 1724, 1808, 1894, 1982, 2072, 2164, 2258, 2354, 2452, 2552, 2654, 2758, 2864, 2972, 3082, 3194, 3308, 3424, 3542, 3662, 3784

Offset: 0

N. J. A. Sloane, Aug 01 2025

Maximum number of regions that can be formed in the plane by drawing n circles (of any size), also maximum number of regions that can be formed on the sphere by drawing n great circles.

It is unfortunate that A014206 (which should have been this sequence) starts 2, 4, 8, 14, 22, 32, 44, 58, 74, 92, ... and has offset 0, but it is much too late to change it now. A014206 is, however, the main entry for this problem and the present sequence has been created to serve as a pointer to it.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 3.

Paolo Xausa, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, 5 circles can divide the plane into a(5) = 22 regions [The circles here are all the same size, although that was not a requirement. The same construction works for any n: take n equally-spaced centers around a circle. Then use inverse stereographic progression to get n great circles on a sphere.]
Eric Weisstein's World of Mathematics, Plane Division by Circles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

See A014206 for further information (including additional references).

A386480[n_] := If[n == 0, 1, n*(n - 1) + 2];
Array[A386480, 100, 0] (* Paolo Xausa, Aug 01 2025 *)

From Stefano Spezia, Aug 01 2025: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
G.f.: (1 - x + x^2 + x^3)/(1 - x)^3.
E.g.f.: exp(x)*(2 + x^2) - 1. (End)
a(n) = A003682(n) = A002061(n)+1, n>=2. - R. J. Mathar, Aug 03 2025

cp's OEIS Frontend

A014206 a(n) = n^2 + n + 2.

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A332307 Array read by antidiagonals: T(m,n) is the number of (undirected) Hamiltonian paths in the m X n grid graph.

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A137882 Number of (directed) Hamiltonian paths in the n-ladder graph.

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A219686 T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, vertical or antidiagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..2 nXk array.

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A221542 T(n,k) = Number of 0..k arrays of length n with each element differing from at least one neighbor by something other than 1, starting with 0.

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A201375 T(n,k)=Number of nXk 0..1 arrays with rows and columns lexicographically nondecreasing read forwards, and nonincreasing read backwards.

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A386480 a(0) = 1; thereafter a(n) = n^2 - n + 2.

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A144336 Triangle read by rows, 2*A144328 - A007318^(-1).

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