OFFSET

0,1

COMMENTS

Draw n + 1 circles in the plane; sequence gives maximal number of regions into which the plane is divided (a(n) = A002061(n+1) + 1 for n >= 0). Cf. A051890.

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)

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.

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)

LINKS

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).

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