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%I #114 Nov 17 2023 11:33:58
%S 2,2,6,14,26,42,62,86,114,146,182,222,266,314,366,422,482,546,614,686,
%T 762,842,926,1014,1106,1202,1302,1406,1514,1626,1742,1862,1986,2114,
%U 2246,2382,2522,2666,2814,2966,3122,3282,3446,3614,3786,3962
%N a(n) = 2*(n^2 - n + 1).
%C 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 (cf. A014206).
%C 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
%C For n > 2, third diagonal of A154685. - _Vincenzo Librandi_, Aug 06 2010
%C a(k) is also the Moore lower bound A198300(k,6) on the order A054760(k,6) of a (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
%C From _Jess Tauber_, May 20 2013: (Start)
%C For neutron shell filling in spherical atomic nuclei, this sequence shows numerical differences between filled spin-split suborbitals sharing all quantum numbers except the principal quantum number n, and here all n's must differ by 1. Only a small handful of exceptions exist.
%C This sequence consists of summed pairs of every other doubled triangular number. It also can be created by taking differences between nuclear magic numbers from the harmonic oscillator (HO)(doubled tetrahedral) set and the spin-orbit (SO) set (2,6,14,28,50,82,126,184,...), with either set being larger. So SO-HO: 2-0=2, 6-0=6, 14-0=14, 28-2=26, 50-8=42, 82-20=62, 126-40=86, 184-70=114, and HO-SO: 2-0=2, 8-2=6, 20-6=14, 40-14=26, 70-28=42, 112-50=62, 168-82=86, 240-126=114. From the perspective of idealized HO periodic structure, with suborbitals in order from largest to smallest spin, alternating by parity, the HO-SO set is spaced two period analogs PLUS one suborbital, while the SO-HO set is spaced two period analogs MINUS one suborbital. (end)
%C The known values of f(k,6) and F(k,6) in Brown (1967), Table 1, closely match this sequence. - _N. J. A. Sloane_, Jul 09 2015
%C Numbers k such that 2*k - 3 is a square. - _Bruno Berselli_, Nov 08 2017
%C Numbers written 222 in number base B, including binary with 'digit' 2: 222(2)=14, 222(3)=26, ... - _Ron Knott_, Nov 14 2017
%H G. C. Greubel, <a href="/A051890/b051890.txt">Table of n, a(n) for n = 0..5000</a>
%H William G. Brown, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.104.9260">On Hamiltonian regular graphs of girth six</a>, J. London Math. Soc., 42 (1967): 514-520.
%H Steven Edwards and William Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Griffiths/griffiths51.html">On Generalized Delannoy Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
%H William Q. Erickson and Jan Kretschmann, <a href="https://arxiv.org/abs/2311.07522">The structure and normalized volume of Monge polytopes</a>, arXiv:2311.07522 [math.CO], 2023. See p. 10.
%H Parabola, <a href="http://www.parabola.unsw.edu.au/1980-1989/volume-20-1984/issue-2/article/problems-section-problems-600-611">Problem #Q607</a>, vol. 20, no. 2, 1984, p. 27.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlaneDivisionbyEllipses.html">Plane Division by Ellipses</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 4*binomial(n, 2) + 2. - Francois Jooste (phukraut(AT)hotmail.com), Mar 05 2003
%F For n > 2, nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^5). - _Benoit Cloitre_, Jun 12 2003
%F a(n) = 2*A002061(n). - _Jonathan Vos Post_, Jun 19 2005
%F a(n) = 4*n + a(n-1) - 4 for n > 0, a(0)=2. - _Vincenzo Librandi_, Aug 06 2010
%F 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
%F G.f.: 2*(1 - 2*x + 3*x^2)/(1 - x)^3. - _Colin Barker_, Jan 10 2012
%F a(n) = A001844(n-1) + 1 = A046092(n-1) + 2. - _Jaroslav Krizek_, Dec 27 2013
%F E.g.f.: 2*(x^2 + 1)*exp(x). - _G. C. Greubel_, Jul 14 2017
%p A051890 := n->2*(n^2-n+1); seq(A051890(n) = n=0..50);
%t Table[2*(n^2-n+1), {n, 0, 50}] (* _G. C. Greubel_, Jul 14 2017 *)
%o (PARI) a(n)=2*(n^2-n+1) \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [2*(n^2-n+1): n in [0..50]]; // _G. C. Greubel_, Feb 21 2019
%o (Sage) [2*(n^2-n+1) for n in (0..50)] # _G. C. Greubel_, Feb 21 2019
%o (GAP) List([0..50], n-> 2*(n^2-n+1)) # _G. C. Greubel_, Feb 21 2019
%Y Cf. A001844, A002061, A014206, A154685, A195600.
%Y 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).
%K nonn,easy
%O 0,1
%A Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Apr 30 2000
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