A000212 a(n) = floor(n^2/3). (Formerly M2439 N0966)

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

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) is the 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 an 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

From Gus Wiseman, Oct 05 2020: (Start)

Also the number of unimodal triples (meaning the middle part is not strictly less than both of the other two) of positive integers summing to n + 1. The a(2) = 1 through a(6) = 12 triples are:

(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5)

(1,2,1) (1,2,2) (1,2,3) (1,2,4)

(2,1,1) (1,3,1) (1,3,2) (1,3,3)

(2,2,1) (1,4,1) (1,4,2)

(3,1,1) (2,2,2) (1,5,1)

(2,3,1) (2,2,3)

(3,2,1) (2,3,2)

(4,1,1) (2,4,1)

(3,2,2)

(3,3,1)

(4,2,1)

(5,1,1)

(End)

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

Kevin Beanland, Hung Viet Chu, and Carrie E. Finch-Smith, Generalized Schreier sets, linear recurrence relation, Turán graphs, arXiv:2112.14905 [math.CO], 2021.

Rafael Durbano Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem.

Bakir Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.

Bakir Farhi, An Elementary Proof that any Natural Number can be Written as the Sum of Three Terms of the Sequence floor(n^2/3), Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.6.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

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

C. K. Wong and Don 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*A049347(n+2-k). - 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

a(n) = Sum_{i = 1..n} floor(2*i/3). - Wesley Ivan Hurt, May 22 2017

a(-n) = a(n). - Paul Curtz, Jan 19 2020

a(n) = A001399(2*n - 3). - Gus Wiseman, Oct 07 2020

a(n) = (1/6)*(2*n^2 - 3 + gcd(n,3)). - Ridouane Oudra, Apr 15 2021

E.g.f.: (exp(x)*(-2 + 3*x*(1 + x)) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 24 2022

Sum_{n>=2} (-1)^n/a(n) = Pi/sqrt(3) - Pi^2/36 - 3/4. - Amiram Eldar, Dec 02 2022

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 + ...
From Gus Wiseman, Oct 07 2020: (Start)
The a(2) = 1 through a(6) = 12 partitions of 2*n into exactly 3 parts (Barker) are the following. The Heinz numbers of these partitions are given by the intersection of A014612 (triples) and A300061 (even sum).
  (2,1,1)  (2,2,2)  (3,3,2)  (4,3,3)  (4,4,4)
           (3,2,1)  (4,2,2)  (4,4,2)  (5,4,3)
           (4,1,1)  (4,3,1)  (5,3,2)  (5,5,2)
                    (5,2,1)  (5,4,1)  (6,3,3)
                    (6,1,1)  (6,2,2)  (6,4,2)
                             (6,3,1)  (6,5,1)
                             (7,2,1)  (7,3,2)
                             (8,1,1)  (7,4,1)
                                      (8,2,2)
                                      (8,3,1)
                                      (9,2,1)
                                      (10,1,1)
(End)

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

Table[Quotient[n^2, 3], {n, 0, 59}] (* 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
(Python)
def A000212(n): return n**2//3 # Chai Wah Wu, Jun 07 2022

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

Cf. A005408.

A000217(n-2) counts 3-part compositions.

A014612 ranks 3-part partitions, with strict case A007304.

A069905 counts the 3-part partitions.

A211540 counts strict 3-part partitions.

A337453 ranks strict 3-part compositions.

Cf. A056834, A130519, A056838, A056865.

A001399(n-6)*4 is the strict version.

A001523 counts unimodal compositions, with strict case A072706.

A001840(n-4) is the non-unimodal version.

A001399(n-6)*2 is the strict non-unimodal version.

A007052 counts unimodal patterns.

A115981 counts non-unimodal compositions, ranked by A335373.

A011782 counts unimodal permutations.

A335373 is the complement of a ranking sequence for unimodal compositions.

A337459 ranks these compositions, with complement A337460.

Cf. A069905, A220377, A332743, A337461, A337561, A337603, A337604.

Sequence in context: A186494 A371702 A194180 * A183139 A094913 A265060

Adjacent sequences: A000209 A000210 A000211 * A000213 A000214 A000215

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Edited by Charles R Greathouse IV, Apr 19 2010

Status

approved