cp's OEIS Frontend

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

%I A014206 #196 Aug 01 2025 13:44:36
%S A014206 2,4,8,14,22,32,44,58,74,92,112,134,158,184,212,242,274,308,344,382,
%T A014206 422,464,508,554,602,652,704,758,814,872,932,994,1058,1124,1192,1262,
%U A014206 1334,1408,1484,1562,1642,1724,1808,1894,1982,2072,2164,2258,2354,2452,2552
%N A014206 a(n) = n^2 + n + 2.
%C A014206 Draw n + 1 circles in the plane; sequence gives maximal number of regions into which the plane is divided. Cf. A051890, A386480.
%C A014206 Number of binary (zero-one) bitonic sequences of length n + 1. - Johan Gade (jgade(AT)diku.dk), Oct 15 2003
%C A014206 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 A014206 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 A014206 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 A014206 A related sequence is A241119. - _Avi Friedlich_, Apr 28 2015
%C A014206 From _Avi Friedlich_, Apr 28 2015: (Start)
%C A014206 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 A014206 a(3*k-3) = 9*k^2 - 15*k + 8,
%C A014206 a(3*k-2) = 9*k^2 - 9*k  + 4,
%C A014206 a(3*k-1) = 9*k^2 - 3*k  + 2,
%C A014206 a(3*k)   = 3*(k+1)^2  - 1. (End)
%C A014206 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 A014206 For prime p and any integer k, k^a(p-1) == k^2 (mod p^2). - _Jianing Song_, Apr 20 2019
%C A014206 From _Bernard Schott_, Jan 01 2021: (Start)
%C A014206 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 A014206 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)
%C A014206 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
%D A014206 K. E. Batcher, Sorting Networks and their Applications. Proc. AFIPS Spring Joint Comput. Conf., Vol. 32, pp. 307-314 (1968). [for bitonic sequences]
%D A014206 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 3.
%D A014206 T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms. MIT Press / McGraw-Hill (1990) [for bitonic sequences]
%D A014206 Indiana School Mathematics Journal, vol. 14, no. 4, 1979, p. 4.
%D A014206 D. E. Knuth, The Art of Computer Programming, vol3: Sorting and Searching, Addison-Wesley (1973) [for bitonic sequences]
%D A014206 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 177.
%D A014206 Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83.
%D A014206 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 A014206 N. J. A. Sloane, <a href="/A014206/b014206.txt">Table of n, a(n) for n = 0..1000</a>
%H A014206 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 A014206 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A014206 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A014206 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 A014206 Hans Werner Lang, <a href="http://www.iti.fh-flensburg.de/lang/algorithmen/sortieren/bitonic/bitonicen.htm">Bitonic sequences</a>.
%H A014206 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 A014206 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 A014206 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 A014206 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 A014206 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 A014206 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 A014206 Yoshio Sano, <a href="http://arxiv.org/abs/0905.1763">The competition numbers of regular polyhedra</a>, arXiv:0905.1763 [math.CO], 2009.
%H A014206 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 A014206 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlaneDivisionbyCircles.html">Plane Division by Circles</a>.
%H A014206 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A014206 G.f.: 2*(x^2 - x + 1)/(1 - x)^3.
%F A014206 n hyperspheres divide R^k into at most C(n-1, k) + Sum_{i = 0..k} C(n, i) regions.
%F A014206 a(n) = A002061(n+1) + 1 for n >= 0. - _Rick L. Shepherd_, May 30 2005
%F A014206 Equals binomial transform of [2, 2, 2, 0, 0, 0, ...]. - _Gary W. Adamson_, Jun 18 2008
%F A014206 a(n) = A003682(n+1), n > 0. - _R. J. Mathar_, Oct 28 2008
%F A014206 a(n) = a(n-1) + 2*n (with a(0) = 2). - _Vincenzo Librandi_, Nov 20 2010
%F A014206 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 A014206 a(n + 1) = n^2 + 3*n + 4. - _Alonso del Arte_, Apr 12 2015
%F A014206 a(n) = Sum_{i=n-2..n+2} i*(i + 1)/5. - _Bruno Berselli_, Oct 20 2016
%F A014206 Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(7)/2)/sqrt(7). - _Amiram Eldar_, Jan 09 2021
%F A014206 From _Amiram Eldar_, Jan 29 2021: (Start)
%F A014206 Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(11)*Pi/2)*sech(sqrt(7)*Pi/2).
%F A014206 Product_{n>=0} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)*sech(sqrt(7)*Pi/2). (End)
%F A014206 a(n) = 2*A000124(n). - _R. J. Mathar_, Mar 14 2021
%F A014206 E.g.f.: exp(x)*(2 + 2*x + x^2). - _Stefano Spezia_, Apr 30 2022
%e A014206 a(0) = 0^2 + 0 + 2 = 2.
%e A014206 a(1) = 1^2 + 1 + 2 = 4.
%e A014206 a(2) = 2^2 + 2 + 2 = 8.
%e A014206 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 A014206 := n->n^2+n+2;
%t A014206 Table[n^2 + n + 2, {n, 0, 50}] (* _Stefan Steinerberger_, Apr 08 2006 *)
%t A014206 LinearRecurrence[{3, -3, 1}, {2, 4, 8}, 50] (* _Harvey P. Dale_, May 14 2011 *)
%t A014206 CoefficientList[Series[2 (x^2 - x + 1)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Apr 29 2015 *)
%o A014206 (PARI) a(n)=n^2+n+2 \\ _Charles R Greathouse IV_, Jul 31 2011
%o A014206 (PARI) x='x+O('x^100); Vec(2*x*(x^2-x+1)/(1-x)^3) \\ _Altug Alkan_, Nov 01 2015
%o A014206 (Magma) [n^2+n+2: n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2015
%Y A014206 Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).
%Y A014206 A row of A059250.
%Y A014206 Cf. A000124, A051890, A002522, A241119, A033547 (partial sums).
%Y A014206 Cf. A002061 (central polygonal numbers).
%Y A014206 Cf. A003682, A160450, A160457, A200182, A386480.
%Y A014206 Column 4 of A347570.
%K A014206 nonn,easy,nice
%O A014206 0,1
%A A014206 _N. J. A. Sloane_
%E A014206 More terms from _Stefan Steinerberger_, Apr 08 2006