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A000212 a(n) = floor(n^2/3).
(Formerly M2439 N0966)
43
0, 0, 1, 3, 5, 8, 12, 16, 21, 27, 33, 40, 48, 56, 65, 75, 85, 96, 108, 120, 133, 147, 161, 176, 192, 208, 225, 243, 261, 280, 300, 320, 341, 363, 385, 408, 432, 456, 481, 507, 533, 560, 588, 616, 645, 675, 705, 736, 768, 800, 833, 867, 901, 936 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Let M_n be the n X n matrix of the following form: [3 2 1 0 0 0 0 0 0 0 / 2 3 2 1 0 0 0 0 0 0 / 1 2 3 2 1 0 0 0 0 0 / 0 1 2 3 2 1 0 0 0 0 / 0 0 1 2 3 2 1 0 0 0 / 0 0 0 1 2 3 2 1 0 0 / 0 0 0 0 1 2 3 2 1 0 / 0 0 0 0 0 1 2 3 2 1 / 0 0 0 0 0 0 1 2 3 2 / 0 0 0 0 0 0 0 1 2 3]. Then for n>2 a(n) = det M_(n-2). - Benoit Cloitre, Jun 20 2002

Largest possible size for the directed Cayley graph on two generators having diameter n-2. - Ralf Stephan, Apr 27 2003

It seems that for n >= 2 a(n) = maximum number of non-overlapping 1 X 3 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's tool, see links. - Dmitry Kamenetsky, Aug 03 2009

Maximum number of edges in a K4-free graph with n vertices. - Yi Yang, May 23 2012

3a(n) + 1 = y^2 if n is not 0 mod 3 and 3a(n) = y^2 otherwise. - Jon Perry, Sep 10 2012

Apart from the initial term this is the elliptic troublemaker sequence R_n(1,3) (also sequence R_n(2,3)) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 08 2013

The number of partitions of 2n into exactly 3 parts. - Colin Barker, Mar 22 2015

a(n-1) is the maximum number of non-overlapping triples (i,k),(i+1,k+1),(i+2,k+2) in a n X n matrix. Details: The triples are distributed along the main diagonal and 2*(n-1) other diagonals. Their maximum number is floor(n/3) + 2*Sum_{k=1..n-1} floor(k/3) = floor((n-1)^2/3). - Gerhard Kirchner, Feb 04 2017

Conjecture:  a(n) is the number of intersection points of n сevians that cut a triangle into the maximum number of pieces (see A007980). - Anton Zakharov, May 07 2017

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

R. D. Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051v3 [math.NT], 2011-2014.

C. K. Wong and D. Coppersmith, A combinatorial problem related to multimodule memory organizations, J. ACM 21 (1974), 392-402.

Anton Zakharov, cevians

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

FORMULA

G.f.: x^2*(1+x)/((1-x)^2*(1-x^3)). - Franklin T. Adams-Watters, Apr 01 2002

Euler transform of length 3 sequence [ 3, -1, 1]. - Michael Somos, Sep 25 2006

G.f.: x^2 * (1 - x^2) / ((1 - x)^3 * (1 - x^3)). a(-n) = a(n). - Michael Somos, Sep 25 2006

a(n) = Sum_{k=0..n} A011655(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009

a(n) = a(n-1) + a(n-3) - a(n-4) + 2 for n>=4. - Alexander Burstein, Nov 20 2011

a(n) = a(n-3) + A005408(n-2) for n>=3. - Alexander Burstein, Feb 15 2013

a(n) = (n-1)^2 - a(n-1) - a(n-2) for n>=2. - Richard R. Forberg, Jun 05 2013

Sum_{n>=2, 1/a(n)} = (27 + 6*sqrt(3)*Pi + 2*Pi^2)/36. - Enrique Pérez Herrero, Jun 29 2013

0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014

a(n) = Sum_{k=1..n} k^2*b(n+2-k), where b(n)=A049347. - Mircea Merca, Feb 04 2014

a(n) = Sum_{i=1..n+1} (ceiling(i/3) + floor(i/3) - 1). - Wesley Ivan Hurt, Jun 06 2014

a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n-1)/3). - Wesley Ivan Hurt, Mar 12 2015

EXAMPLE

G.f. = x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 21*x^8 + 27*x^9 + 33*x^10 + ...

MAPLE

A000212:=(-1+z-2*z**2+z**3-2*z**4+z**5)/(z**2+z+1)/(z-1)**3; # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence with an additional leading 1.

A000212 := proc(n) option remember; `if`(n<4, [0, 0, 1, 3][n+1], a(n-1)+a(n-3) -a(n-4)+2) end; # Peter Luschny, Nov 20 2011

MATHEMATICA

k0=k1=0; lst={k0, k1}; Do[kt=k1; k1=n^2-k1-k0; k0=kt; AppendTo[lst, k1], {n, 1, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)

a[ n_] := Quotient[ n^2, 3] (* Michael Somos, Jan 22 2014 *)

PROG

(PARI) {a(n) = n^2 \ 3}; /* Michael Somos, Sep 25 2006 */

(MAGMA) [Floor(n^2 / 3): n in [0..50]]; // Vincenzo Librandi, May 08 2011

CROSSREFS

Cf. A000290, A007590 (= R_n(2,4)), A002620 (= R_n(1,2)), A118015, A056827, A118013.

Cf. A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A033440, A033441, A033442, A033443, A033444.

Cf. A001353 and A004523 (first differences). A184535 (= R_n(2,5) = R_n(3,5)).

Cf. A238738. [Bruno Berselli, Apr 17 2015]

Cf. A007980

Sequence in context: A194176 A186494 A194180 * A183139 A094913 A265060

Adjacent sequences:  A000209 A000210 A000211 * A000213 A000214 A000215

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Apr 30 1991

EXTENSIONS

Edited by Charles R Greathouse IV, Apr 19 2010

STATUS

approved

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Last modified May 22 19:31 EDT 2017. Contains 286885 sequences.