OFFSET
0,1
COMMENTS
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
REFERENCES
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.
Jacob Steiner, Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math., 1 (1826), 349-364. See Eq. (11). [Note that the title uses the old spelling of Teilung. This is not a typo. - N. J. A. Sloane, Mar 30 2026]
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; solutions pp. 102-107 (First published: San Francisco: Holden-Day, Inc., 1964).
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..1000
Alexander Burstein, Sergey Kitaev, and Toufik Mansour, Partially ordered patterns and their combinatorial interpretations, P. U. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.
David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Suh-Ryung Kim and Yoshio Sano, The competition numbers of complete tripartite graphs, Discrete Appl. Math., 156 (2008) 3522-3524.
Hans Werner Lang, Bitonic sequences.
Adrián Lillo and Mercedes Rosas, The Priority Lattice, arXiv:2603.28905 [math.CO], 2026. See p. 24.
Daniel Q. Naiman and Edward R. Scheinerman, Arbitrage and Geometry, arXiv:1709.07446 [q-fin.MF], 2017.
Johan Nilsson, The Pattern Complexity of the Sierpiński Triangle, arXiv:2510.06493 [math.CO], 2025. See p. 2.
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.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I, Annotated scan of pp. 102-103.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I, Annotated scan of pp. 104-105.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I, Annotated scan of pp. 106-107.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
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
EXAMPLE
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
MAPLE
A014206 := n->n^2+n+2;
MATHEMATICA
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 *)
PROG
(PARI) a(n)=n^2+n+2 \\ Charles R Greathouse IV, Jul 31 2011
(PARI) x='x+O('x^100); Vec(2*x*(x^2-x+1)/(1-x)^3) \\ Altug Alkan, Nov 01 2015
(Magma) [n^2+n+2: n in [0..50]]; // Vincenzo Librandi, Apr 29 2015
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Stefan Steinerberger, Apr 08 2006
STATUS
approved
