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2, 2, 6, 14, 26, 42, 62, 86, 114, 146, 182, 222, 266, 314, 366, 422, 482, 546, 614, 686, 762, 842, 926, 1014, 1106, 1202, 1302, 1406, 1514, 1626, 1742, 1862, 1986, 2114, 2246, 2382, 2522, 2666, 2814, 2966, 3122, 3282, 3446, 3614, 3786, 3962
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history;
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OFFSET
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0,1
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COMMENTS
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Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; sequence gives number of regions into which the plane is divided.
Least k such that Z(k,2) <= Z(n,3) where Z(m,s) = sum(i>=m, 1/i^s) = zeta(s)-sum(i=1,m-1,1/i^s). - Benoit Cloitre, Nov 29 2002
For n>2, third diagonal of [A154685] [From Vincenzo Librandi, Aug 06 2010]
a(k) is also the Moore lower bound A198300(k,6) on the order A054760(k,6) of an (k,6)-cage. Equality is achieved if and only if there exists a finite projective plane of order k - 1. A sufficient condition for this is that k - 1 be a prime power. - Jason Kimberley, Oct 17 2011 and Jan 01 2013
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REFERENCES
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Parabola, vol. 20, no. 2, 1984, p. 27, Problem #Q607.
J. V. Post, "When Centered Polygonal Numbers are Perfect Squares" preprint.
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LINKS
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Table of n, a(n) for n=0..45.
Parabola, Web site
Eric Weisstein's World of Mathematics, Plane Division by Ellipses
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 4*binomial(n, 2)+2. - Francois Jooste (phukraut(AT)hotmail.com), Mar 05 2003
For n>2 nearest integer to sum(k>=n, 1/k^3)/sum(k>=n, 1/k^5) - Benoit Cloitre, Jun 12 2003
a(n) = 2*A002061(n). - Jonathan Vos Post, Jun 19 2005
a(n) = 4*n+a(n-1)-4 (with a(0)=2) [From Vincenzo Librandi, Aug 06 2010]
a(n) = 2*(n^2-n+1) = 2*(n-1)^2 + 2(n-1) + 2 = 222 read in base n-1 (for n>3). - Jason Kimberley, Oct 20 2011
G.f.: 2*(1-2*x+3*x^2)/(1-x)^3. [Colin Barker, Jan 10 2012]
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EXAMPLE
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a(1)=4*1+2-4=2; a(2)=4*2+2-4=6; a(3)=4*3+6-4=14 [From Vincenzo Librandi, Aug 06 2010]
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MAPLE
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A051890 := n->2*(n^2-n+1);
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MATHEMATICA
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a=2; lst={}; Do[a+=n; AppendTo[lst, a], {n, 0, 6!, 4}]; lst...and/or... lst={}; Do[AppendTo[lst, 2*(n^2-n+1)], {n, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Mar 01 2009]
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CROSSREFS
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Cf. A001844, A002061, A014206, A002061.
Cf. A154685 [From Vincenzo Librandi, Jan 25 2009]
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), this sequence (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Sequence in context: A049952 A019100 A019101 * A071109 A005310 A002203
Adjacent sequences: A051887 A051888 A051889 * A051891 A051892 A051893
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KEYWORD
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nonn,easy
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AUTHOR
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Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Apr 30 2000
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STATUS
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approved
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