0,1

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).

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 (zabrocki(AT)mathstat.yorku.ca), 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 R. Janjic (agnus(AT)blic.net), Dec 28 2007

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.

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]

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.

S.-R. Kim and Y. Sano: The competition numbers of complete tripartite graphs, Discrete Appl. Math., 156 (2008) 3522-3524.

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.

J. C. Novelli and A. Schilling, The Forgotten Moniod, http://arXiv.org/abs/0706.2996

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)

N. J. A. Sloane, Table of n, a(n) for n = 0..1000

Author? Bitonic sequences

Parabola, vol. 24, no. 1, 1988, p. 22, Problem #Q736.

Yoshio Sano, The competition numbers of regular polyhedra, May 12, 2009.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to linear recurrences with constant coefficients

G.f.: 2*x*(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 (rshepherd2(AT)hotmail.com), May 30 2005

((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

Equals binomial transform of [2, 2, 2, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 18 2008

a(n)=A003682(n+1), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 28 2008]

If a(1)=2 then a(n+1)=a(n)+2n [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jul 20 2009]

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]

a(0) = 0^2+0+2 = 2, a(1) = 1^2+1+2 =4, a(2) = 2^2+2+2 = 8, etc.

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]

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]

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

with (combinat):seq(fibonacci(3, n)+n+1, n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2008

Table[n^2 + n + 2, {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006

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

Cf. A000124, A051890. Also A033547=partial sums of A014206.

Cf. A002061 (Central polygonal numbers).

Cf. A002522.

Sequence in context: A132425 A154264 A155506 this_sequence A025196 A084626 A090533

Adjacent sequences: A014203 A014204 A014205 this_sequence A014207 A014208 A014209

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006