A000212 a(n) = floor(n^2/3).
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
Examples
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)
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).
Crossrefs
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. A238738. - Bruno Berselli, Apr 17 2015
Cf. A007980
Cf. A005408.
A000217(n-2) counts 3-part compositions.
A069905 counts the 3-part partitions.
A211540 counts strict 3-part partitions.
A337453 ranks strict 3-part compositions.
A001399(n-6)*4 is the strict version.
A001840(n-4) is the non-unimodal version.
A001399(n-6)*2 is the strict non-unimodal version.
A007052 counts unimodal patterns.
A011782 counts unimodal permutations.
A335373 is the complement of a ranking sequence for unimodal compositions.
Programs
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Magma
[Floor(n^2 / 3): n in [0..50]]; // Vincenzo Librandi, May 08 2011
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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
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Mathematica
Table[Quotient[n^2, 3], {n, 0, 59}] (* Michael Somos, Jan 22 2014 *) -
PARI
{a(n) = n^2 \ 3}; /* Michael Somos, Sep 25 2006 */ -
Python
def A000212(n): return n**2//3 # Chai Wah Wu, Jun 07 2022
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
Extensions
Edited by Charles R Greathouse IV, Apr 19 2010
Comments