%I A014206
%S A014206 2,4,8,14,22,32,44,58,74,92,112,134,158,184,212,242,274,308,344,382,422,
%T A014206 464,508,554,602,652,704,758,814,872,932,994,1058,1124,1192,1262,1334,
%U A014206 1408,1484,1562,1642,1724,1808,1894,1982,2072,2164,2258,2354,2452,2552
%N A014206 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 (a(n) = A002061(n+1) + 1 for n>=0).
%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 (zabrocki(AT)mathstat.yorku.ca), 
               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 R. Janjic (agnus(AT)blic.net), 
               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 (jvospost3(AT)gmail.com), 
               May 14 2009. Cf. A160450, A160457.
%C A014206 Except for the first term, a(n)=2*n+a(n-1), (with a(1)=4) [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Oct 23 2009]
%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 S.-R. Kim and Y. Sano: The competition numbers of complete tripartite 
               graphs, Discrete Appl. Math., 156 (2008) 3522-3524.
%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 J. C. Novelli and A. Schilling, The Forgotten Moniod, http://arXiv.org/
               abs/0706.2996
%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="b014206.txt">Table of n, a(n) for n = 0..1000</
               a>
%H A014206 Author? <a href="http://www.iti.fh-flensburg.de/lang/algorithmen/sortieren/
               bitonic/bitonicen.htm">Bitonic sequences</a>
%H A014206 Parabola, <a href="http://www.maths.unsw.edu.au/Parabola/">vol. 24, no. 
               1, 1988, p. 22, Problem #Q736.</a>
%H A014206 Yoshio Sano, <a href="http://arxiv.org/abs/0905.1763">The competition 
               numbers of regular polyhedra</a>, May 12, 2009.
%H A014206 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PlaneDivisionbyCircles.html">Link to a section of The World of Mathematics.</
               a>
%H A014206 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A014206 G.f.: 2*x*(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 (rshepherd2(AT)hotmail.com), 
               May 30 2005
%F A014206 ((binomial(n+3,n+1)-binomial(n+1,n))*(binomial(n+3,n+2)-binomial(n+1,
               n)). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 12 2006
%F A014206 Equals binomial transform of [2, 2, 2, 0, 0, 0,...]. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jun 18 2008
%F A014206 a(n)=A003682(n+1), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 28 2008]
%F A014206 If a(1)=2 then a(n+1)=a(n)+2n [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Jul 20 2009]
%F A014206 Except for the first term, a(n)=2*n+a(n-1), (with a(1)=4) [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Oct 23 2009]
%e A014206 a(0) = 0^2+0+2 = 2, a(1) = 1^2+1+2 =4, a(2) = 2^2+2+2 = 8, etc.
%e A014206 For n=2, a(2)=2+2=4; n=3, a(2+1)=a(2)+2n=4+4=8 [From Vincenzo Librandi 
               (vincenzo.librandi(AT)tin.it), Jul 20 2009]
%e A014206 For n=2, a(2)=2*2+4=8; n=3, a(3)=2*3+8=14; n=4, a(4)=2*4+14=22 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 23 2009]
%p A014206 A014206 := n->n^2+n+2;
%p A014206 with (combinat):seq(fibonacci(3, n)+n+1, n=0..50); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 07 2008
%t A014206 Table[n^2 + n + 2, {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 08 2006
%Y A014206 Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5). 
               A row of A059250.
%Y A014206 Cf. A000124, A051890. Also A033547=partial sums of A014206.
%Y A014206 Cf. A002061 (Central polygonal numbers).
%Y A014206 Cf. A002522.
%Y A014206 Sequence in context: A132425 A154264 A155506 this_sequence A025196 A084626 
               A090533
%Y A014206 Adjacent sequences: A014203 A014204 A014205 this_sequence A014207 A014208 
               A014209
%K A014206 nonn,easy,nice
%O A014206 0,1
%A A014206 N. J. A. Sloane (njas(AT)research.att.com).
%E A014206 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 08 2006