A032765 a(n) = floor(n*(n+1)*(n+2) / (n + n+1 + n+2)), which equals floor(n*(n+2)/3).

0, 1, 2, 5, 8, 11, 16, 21, 26, 33, 40, 47, 56, 65, 74, 85, 96, 107, 120, 133, 146, 161, 176, 191, 208, 225, 242, 261, 280, 299, 320, 341, 362, 385, 408, 431, 456, 481, 506, 533, 560, 587, 616, 645, 674, 705, 736, 767, 800, 833, 866, 901, 936, 971, 1008

Offset

0,3

Comments

Satisfies a(n+1) - 2*a(n) + a(n-1) = (2/3)*(1 + w^(n+1) + w^(2n+2)), a(0)=0 & a(1)=1 where w is the imaginary cubic root of unity. - Robert G. Wilson v, Jun 24 2002

First differences have this pattern: (+1) +1 +1 +3 +3 +3 +5 +5 +5 +7 +7 +7 +9 +9 +9. - Alexandre Wajnberg, Dec 19 2005

In Duistermaat (2010) section 11.3 The Planar Four-Bar Link on page 516: "It follows from (4.3.2) that the number of k-periodic fibers of the QRT automorphism, counted with multiplicities, is equal to \nu(\tau^S)^k) = 3n^2 - 1 when k = 3n, 3n^2 + 2n when k = 3n + 1, 3n^2 + 4n + 1 when k = 3n + 2, for every integer n." - Michael Somos, Mar 14 2023

References

J. J. Duistermaat, Discrete Integrable Systems, 2010, Springer Science+Business Media.

Links

Table of n, a(n) for n=0..54.

J. Bader, Kobon Triangles

Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007

Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007 [Cached copy, with permission]

Taylor Short, The saturation number of carbon nanocones and nanotubes, arXiv:1807.11355 [math.CO], 2018.

Eric Weisstein's World of Mathematics, Kobon Triangle

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

Formula

From Ralf Stephan, May 05 2004: (Start)

a(n) = n^2 - ceiling(n*(n-1)/3).

G.f.: x*(1+2x^2-x^3)/((1+x+x^2)(1-x)^3). (End)

a(n) = floor(n*(n+2)/3). - Saburo Tamura, sent by Alexandre Wajnberg, Dec 19 2005

a(3*n - 1) = 3*n^2 - 1, a(3*n) = 3*n^2 + 2*n, a(3*n + 1) = 3*n^2 + 4*n + 1. - Michael Somos, Mar 14 2023

Example

G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 11*x^5 + 16*x^6 + 21*x^7 + 26*x^8 + ... - Michael Somos, Mar 14 2023

Maple

A032765:=n->floor(n*(n+2)/3); seq(A032765(n), n=0..100); # Wesley Ivan Hurt, Dec 20 2013

Mathematica

Table[ Floor[ n(n + 1)(n + 2)/(n + (n + 1) + (n + 2))], {n, 0, 55}]
Table[Floor[n (n + 2)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 20 2013 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 2, 5, 8}, 60] (* Harvey P. Dale, Jun 06 2016 *)

Prog

(PARI) a(n)=n*(n+2)\3 \\ Charles R Greathouse IV, Jun 11 2015

Crossrefs

Cf. A001082, A032766.

Sequence in context: A361684 A130258 A186496 * A340931 A261416 A340386

Adjacent sequences: A032762 A032763 A032764 * A032766 A032767 A032768

Keyword

nonn,easy

Author

Patrick De Geest, May 15 1998

Extensions

Name change suggested by Wesley Ivan Hurt, Dec 20 2013

Status

approved