A003035 -id:A003035 - cp's OEIS frontend

A058212 a(n) = 1 + floor(n*(n-3)/6).

1, 0, 0, 1, 1, 2, 4, 5, 7, 10, 12, 15, 19, 22, 26, 31, 35, 40, 46, 51, 57, 64, 70, 77, 85, 92, 100, 109, 117, 126, 136, 145, 155, 166, 176, 187, 199, 210, 222, 235, 247, 260, 274, 287, 301, 316, 330, 345, 361, 376, 392, 409, 425, 442, 460, 477, 495, 514, 532, 551

Offset: 0

N. J. A. Sloane, Nov 30 2000

For n >= 3, number of solutions to x+y+z == 0 (mod n) with 0 <= x < y < z < n. E.g., for n=3 there is a unique solution, x=0, y=1, z=2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
S. A. Burr, B. Grünbaum, and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A003035.
Apart from initial term, same as A007997.
The third diagonal of A061857.

a058212 n = 1 + n * (n - 3) `div` 6  -- Reinhard Zumkeller, May 08 2012

Table[Floor[(n(n-3))/6]+1,{n,0,70}] (* or *) LinearRecurrence[{2,-1,1,-2,1},{1,0,0,1,1},70] (* Harvey P. Dale, Jun 21 2021 *)

a(n)=n*(n-3)\6 + 1 \\ Charles R Greathouse IV, Jun 11 2015

[ceil(binomial(n,2)/3) for n in range(-1, 55)] # Zerinvary Lajos, Dec 03 2009

From Paul Barry, Mar 18 2004: (Start)
G.f.: (1 - 2x + x^2 + x^4)/((1 - x)^2(1 - x^3)).
a(n) = 4*cos(2*Pi*n/3)/9 + (3*n^2 - 9*n + 10)/18. (End)
E.g.f.: (exp(x)*(10 - 6*x + 3*x^2) + 8*exp(-x/2)*cos(sqrt(3)*x/2))/18. - Stefano Spezia, May 03 2023
Sum_{n>=3} 1/a(n) = 6 - (2*Pi/sqrt(3))*(1 - tanh(sqrt(5/3)*Pi/2)/sqrt(5)). - Amiram Eldar, May 06 2023

Zerinvary Lajos, Dec 07 2009

A006065 Maximal number of 4-tree rows in n-tree orchard problem.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 5, 6, 7, 9, 10, 12, 15, 16, 18, 20, 23

Offset: 1

N. J. A. Sloane

Maximum number of rows with exactly 4 trees in each row if there are n trees in the orchard.

For further references and links see A003035.

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, Chap. 22.
F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
Xianzu Lin, A new result about orchard-planting problem, Preprint, 2005. [Shows a(20) >= 23.]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
For further references and links see A003035.

P. Berloquin, a(12) >= 7 (from an article in Jeux & Strategies from 1983 - see Fig. 10).
Thomas Bloom, Problem 669 and possibly Problem 101, Erdős Problems.
Stefan A. Burr, Branko Grünbaum, and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
Stefan A. Burr, Branko Grünbaum, and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
Zhao Hui Du, Code to verify a(13) to a(16) for orchard planting problem
Zhao Hui Du, Full list of the optimal results from 13~18 trees
Zhao Hui Du, A Chinese webpage for the problem
Zhao Hui Du, Illustration showing that a(22)>=28 [Line ABCV is infinity line]
Noam D. Elkies, On some points-and-lines problems and configurations, arXiv:math/0612749 [math.MG], 2006.
Erich Friedman, Table of values and bounds for up to 25 trees
Branko Grünbaum and J. F. Rigby, The real configuration (21_4), Journal of the London Mathematical Society 2.2 (1990): 336-346. [Shows a(21) >= 21.]
Xianzu Lin, Illustration showing that a(20) >= 23 [The points S and T are at infinity]
Ed Pegg, Jr., Cultivating New Solutions for theOrchard-Planting Problem, 2018.
Ed Pegg, Jr., Mathpuzzxle Blog, Updated Feb 27 2020. [Gives new construction for n = 22]
Ed Pegg, Jr., Mathpuzzxle Blog, Updated Feb 27 2020. [Gives new construction for n = 22] (extract, local copy)
József Solymosi and Miloš Stojaković, Many collinear k-tuples with no k + 1 collinear points, Discrete & Computational Geometry, October 2013, Volume 50, Issue 3, pp. 811-820; also arXiv 1107.0327 [math.CO], 2011-2013.
Terence Tao, Erdős problem database, see nos. 101, 669.
Eric Weisstein's World of Mathematics, Orchard-Planting Problem.

Cf. A003035, A008997.

Cf. A172992 (the same problem, but with integer-valued tree coordinates).

a(n) >= A172992(n).

a(13)-a(15) from Zhao Hui Du, Aug 24 2008
a(17) from Zhao Hui Du, Nov 11 2008
a(18) from Zhao Hui Du, Nov 25 2008
a(19) from Zhao Hui Du, Dec 17 2009
a(20) from Zhao Hui Du, Feb 01 2010

A008997 Orchard problem with 5 trees in a row (may not have all been proved optimal).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 10, 11

Offset: 1

N. J. A. Sloane

Berloquin shows that a(21) >= 12.

S. A. Burr, B. Grünbaum and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, Chap. 22.

P. Berloquin, a(21) >= 12 (from an article in Jeux & Strategies from 1983 - see Fig. 13)
S. A. Burr, B. Grünbaum and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
Erich Friedman, Table of values and bounds for up to 21 trees
Eric Weisstein's World of Mathematics, Orchard-Planting Problem

Cf. A003035, A006065.

A337747 Maximal number of 4-point circles passing through n points on a plane.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 12, 14, 22, 30, 45

Offset: 1

Dmitry Kamenetsky, Sep 17 2020

This is a variant of the orchard-planting problem that uses circles instead of straight lines.
The maximal number of 3-point circles passing through n points on a plane is binomial(n,3). Given an arrangement of n points in general position, any choice of three points defines a circle. - Peter Kagey, Oct 05 2020
Paul Panzer provides upper and lower bounds:
  a(n) <= floor(n*(n-1)*(n-2)/24).
  a(n) >= 2 + n*((n-2)*(n-2) + 4)/32 for n == 0 (mod 4) and n >= 8.
  a(n) >= 2 + (n-1)*((n-1)*(n-5) + 16)/32 for n == 1 (mod 4) and n >= 9.
  a(n) >= 2 + n*(n-2)*(n-2)/32 for n == 2 (mod 4) and n >= 10.
  a(n) >= 2 + (n-1)*((n-3)*(n-3) + 16)/32 for n == 3 (mod 4) and n >= 11.
It seems that a(n) = n*((n-2)*(n-2) + 4)/32 + 2*A008610(n/2-4) if n == 0 (mod 4) and n >= 8. - Zhao Hui Du, Dec 14 2022
The number of 4-point circles passing through n points (2*cos(t_k), sin(t_k)) where t_k = (2k-1)*Pi/n, k=1,2,...,n is A008610(n-4), so A337747(n) >= A008610(n-4), so A337747(n) ~ n^3/24 for sufficiently large n. - Zhao Hui Du, Dec 15 2022

			See examples in links.

Zhao Hui Du, Source code and running data to show there are at most 30 circles for 11 trees.
Zhao Hui Du, A group of solutions with 30 circles for 11 trees.
Zhao Hui Du, Geogebra dynamic graph for a group of solution to 12 trees 45 circles (4 trees per circle).
Zhao Hui Du, Best known solution for 14 trees (73 circles), removing point I from the solution to form a solution with 13 trees and 53 circles so that a(13) >= 53, a(14) >= 73.
Zhao Hui Du, Graph for 14 trees 73 circles (all black circles have same patterns as that in that in two co-center regular polygons (Paul Panzer's solution) while red circles are those extra circles).
Zhao Hui Du, Graph for 16 trees with 120 circles.
Zhao Hui Du, Geogebra graph for 16 trees with 120 circles (parameter d and m could be any real value and point PP could be moving too), removing any tree to form 15 trees with 90 circles so that a[15]>=90, a[16]>=120.
Zhao Hui Du, Chinese webpage to show that the number of 4 points circle through the n points in ellipse could be transformed to the number to pick 4 different numbers k1,k2,k3,k4 from 1 to n so that n|2+k1+k2+k3+k4, which is equal to A008610(n-4).
Dmitry Kamenetsky, Best known solutions for n <= 13.
Dmitry Kamenetsky, Orchard planting problem, Puzzling StackExchange, August 2020.
Dmitry Kamenetsky, General orchard planting problem, Puzzling StackExchange, September 2020.

Cf. A003035 (the original orchard problem), A006065.

a(11) from Zhao Hui Du, Nov 22 2022

a(12) from Zhao Hui Du, Dec 01 2022

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A058212 a(n) = 1 + floor(n*(n-3)/6).

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A006065 Maximal number of 4-tree rows in n-tree orchard problem.

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A008997 Orchard problem with 5 trees in a row (may not have all been proved optimal).

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A337747 Maximal number of 4-point circles passing through n points on a plane.

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