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A014206 a(n) = n^2 + n + 2. 43

%I

%S 2,4,8,14,22,32,44,58,74,92,112,134,158,184,212,242,274,308,344,382,

%T 422,464,508,554,602,652,704,758,814,872,932,994,1058,1124,1192,1262,

%U 1334,1408,1484,1562,1642,1724,1808,1894,1982,2072,2164,2258,2354,2452,2552

%N a(n) = n^2 + n + 2.

%C 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.

%C Number of binary (zero-one) bitonic sequences of length n + 1. - Johan Gade (jgade(AT)diku.dk), Oct 15 2003

%C 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

%C 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

%C 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.

%C A related sequence is A241119. - _Avi Friedlich_, Apr 28 2015

%C From _Avi Friedlich_, Apr 28 2015: (Start)

%C 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:

%C a(3*k-3) = 9*k^2 - 15*k + 8,

%C a(3*k-2) = 9*k^2 - 9*k + 4,

%C a(3*k-1) = 9*k^2 - 3*k + 2,

%C a(3*k) = 3*(k+1)^2 - 1. (End)

%C 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

%C For prime p and any integer k, k^a(p-1) == k^2 (mod p^2). - _Jianing Song_, Apr 20 2019

%C From _Bernard Schott_, Jan 01 2021: (Start)

%C 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.

%C 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)

%D K. E. Batcher, Sorting Networks and their Applications. Proc. AFIPS Spring Joint Comput. Conf., Vol. 32, pp. 307-314 (1968). [for bitonic sequences]

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

%D T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms. MIT Press / McGraw-Hill (1990) [for bitonic sequences]

%D Indiana School Mathematics Journal, vol. 14, no. 4, 1979, p. 4.

%D D. E. Knuth, The Art of Computer Programming, vol3: Sorting and Searching, Addison-Wesley (1973) [for bitonic sequences]

%D J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 177.

%D Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83.

%D 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)

%H N. J. A. Sloane, <a href="/A014206/b014206.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Burstein, S. Kitaev, and T. Mansour, <a href="http://puma.dimai.unifi.it/19_2_3/3.pdf">Partially ordered patterns and their combinatorial interpretations</a>, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H S.-R. Kim and Y. Sano, <a href="http://dx.doi.org/10.1016/j.dam.2008.04.009">The competition numbers of complete tripartite graphs</a>, Discrete Appl. Math., 156 (2008) 3522-3524.

%H Hans Werner Lang, <a href="http://www.iti.fh-flensburg.de/lang/algorithmen/sortieren/bitonic/bitonicen.htm">Bitonic sequences</a>.

%H Daniel Q. Naiman and Edward R. Scheinerman, <a href="https://arxiv.org/abs/1709.07446">Arbitrage and Geometry</a>, arXiv:1709.07446 [q-fin.MF], 2017.

%H Jean-Christoph Novelli and Anne Schilling, <a href="http://arXiv.org/abs/0706.2996">The Forgotten Monoid</a>, arXiv 0706.2996 [math.CO], 2007.

%H Parabola, <a href="https://www.parabola.unsw.edu.au/files/articles/1980-1989/volume-24-1988/issue-1/vol24_no1_p.pdf">Problem #Q736</a>, 24(1) (1988), p. 22.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1712.06543">Enumerating the states of the twist knot</a>, arXiv:1712.06543 [math.CO], 2017.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1902.08989">A generating polynomial for the two-bridge knot with Conway's notation C(n,r)</a>, arXiv:1902.08989 [math.CO], 2019.

%H Yoshio Sano, <a href="http://arxiv.org/abs/0905.1763">The competition numbers of regular polyhedra</a>, arXiv:0905.1763 [math.CO], 2009.

%H Jeffrey Shallit, <a href="http://recursed.blogspot.com/2012/10">Recursivity: An Interesting but Little-Known Function</a>, 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.]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlaneDivisionbyCircles.html">Plane Division by Circles</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: 2*(x^2 - x + 1)/(1 - x)^3.

%F n hyperspheres divide R^k into at most C(n-1, k) + Sum_{i = 0..k} C(n, i) regions.

%F a(n) = A002061(n+1) + 1 for n >= 0. - _Rick L. Shepherd_, May 30 2005

%F Equals binomial transform of [2, 2, 2, 0, 0, 0, ...]. - _Gary W. Adamson_, Jun 18 2008

%F a(n) = A003682(n+1), n > 0. - _R. J. Mathar_, Oct 28 2008

%F a(n) = a(n-1) + 2*n (with a(0) = 2). - _Vincenzo Librandi_, Nov 20 2010

%F 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

%F a(n + 1) = n^2 + 3*n + 4. - _Alonso del Arte_, Apr 12 2015

%F a(n) = Sum_{i=n-2..n+2} i*(i + 1)/5. - _Bruno Berselli_, Oct 20 2016

%F Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(7)/2)/sqrt(7). - _Amiram Eldar_, Jan 09 2021

%F From _Amiram Eldar_, Jan 29 2021: (Start)

%F Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(11)*Pi/2)*sech(sqrt(7)*Pi/2).

%F Product_{n>=0} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)*sech(sqrt(7)*Pi/2). (End)

%F a(n) = 2*A000124(n). - _R. J. Mathar_, Mar 14 2021

%e a(0) = 0^2 + 0 + 2 = 2.

%e a(1) = 1^2 + 1 + 2 = 4.

%e a(2) = 2^2 + 2 + 2 = 8.

%e a(6) = 4*5/5 + 5*6/5 + 6*7/5 + 7*8/5 + 8*9/5 = 44. - _Bruno Berselli_, Oct 20 2016

%p A014206 := n->n^2+n+2;

%t Table[n^2 + n + 2, {n, 0, 50}] (* _Stefan Steinerberger_, Apr 08 2006 *)

%t LinearRecurrence[{3, -3, 1}, {2, 4, 8}, 50] (* _Harvey P. Dale_, May 14 2011 *)

%t CoefficientList[Series[2 (x^2 - x + 1)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Apr 29 2015 *)

%o (PARI) a(n)=n^2+n+2 \\ _Charles R Greathouse IV_, Jul 31 2011

%o (PARI) x='x+O('x^100); Vec(2*x*(x^2-x+1)/(1-x)^3) \\ _Altug Alkan_, Nov 01 2015

%o (MAGMA) [n^2+n+2: n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2015

%Y Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5). A row of A059250.

%Y Cf. A000124, A051890, A002522, A241119, A033547 (partial sums).

%Y Cf. A002061 (central polygonal numbers).

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_

%E More terms from _Stefan Steinerberger_, Apr 08 2006

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Last modified April 16 03:59 EDT 2021. Contains 343030 sequences. (Running on oeis4.)