A051890 a(n) = 2*(n^2 - n + 1).

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

Offset: 0

Author

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Apr 30 2000

Keywords

• nonn
• easy
• changed

Comments

Draw n ellipses in the plane (n > 0); sequence gives maximum number of regions into which the plane is divided (cf. A014206, A386480).

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. - Vincenzo Librandi, Aug 06 2010

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

From Jess Tauber, May 20 2013: (Start)

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.

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)

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

Numbers k such that 2*k - 3 is a square. - Bruno Berselli, Nov 08 2017

Numbers written 222 in number base B, including binary with 'digit' 2: 222(2)=14, 222(3)=26, ... - Ron Knott, Nov 14 2017

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• William G. Brown, On Hamiltonian regular graphs of girth six, J. London Math. Soc., 42 (1967): 514-520.
• Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
• William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 10.
• Parabola, Problem #Q607, vol. 20, no. 2, 1984, p. 27.
• Eric Weisstein's World of Mathematics, Plane Division by Ellipses
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001844, A002061, A014206, A154685, A195600, A386480.

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).

Programs

GAP

List([0..50], n-> 2*(n^2-n+1)); # G. C. Greubel, Feb 21 2019

Magma

[2*(n^2-n+1): n in [0..50]]; // G. C. Greubel, Feb 21 2019

Maple

A051890 := n->2*(n^2-n+1); seq(A051890(n) = n=0..50);

Mathematica

Table[2*(n^2-n+1), {n, 0, 50}] (* G. C. Greubel, Jul 14 2017 *)
LinearRecurrence[{3,-3,1},{2,2,6},50] (* Harvey P. Dale, Jul 14 2025 *)

PARI

a(n)=2*(n^2-n+1) \\ Charles R Greathouse IV, Sep 24 2015

Sage

[2*(n^2-n+1) for n in (0..50)] # G. C. Greubel, Feb 21 2019

Formula

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 for n > 0, a(0)=2. - 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

a(n) = A001844(n-1) + 1 = A046092(n-1) + 2. - Jaroslav Krizek, Dec 27 2013

E.g.f.: 2*(x^2 + 1)*exp(x). - G. C. Greubel, Jul 14 2017