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2, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158, 184, 212, 242, 274, 308, 344, 382, 422, 464, 508, 554, 602, 652, 704, 758, 814, 872, 932, 994, 1058, 1124, 1192, 1262, 1334, 1408, 1484, 1562, 1642, 1724, 1808, 1894, 1982, 2072, 2164, 2258, 2354, 2452, 2552
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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).
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.
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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.
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
Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83. [From Robert G. Wilson v (rgwv(AT)rgwv.com), May 21 2010]
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)
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..1000
Author? Bitonic sequences
Guo-Niu Han, Enumeration of Standard Puzzles
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, Plane Division by Circles
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| 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]
a(n)=a(n-1)+2*n (with a(0)=2) [From 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) [From Harvey P. Dale, May 14 2011]
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EXAMPLE
| a(0) = 0^2+0+2 = 2, a(1) = 1^2+1+2 =4, a(2) = 2^2+2+2 = 8, etc.
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MAPLE
| 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
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MATHEMATICA
| Table[n^2 + n + 2, {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006
LinearRecurrence[{3, -3, 1}, {2, 4, 8}, 50] (* From Harvey P. Dale, May 14 2011 *)
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PROG
| (PARI) a(n)=n^2+n+2 \\ Charles R Greathouse IV, Jul 31 2011
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CROSSREFS
| 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: A194694 A194692 A155506 * A025196 A084626 A090533
Adjacent sequences: A014203 A014204 A014205 * A014207 A014208 A014209
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 08 2006
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