Sascha Kurz - cp's OEIS frontend

A348469 Maximal number of squares that can be formed from the grid points in a qualifying circular region of the plane that contains exactly n points of a square grid.

0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 8, 11, 13, 15, 17, 20, 22, 25, 28, 32, 37, 40, 43, 47, 51, 56, 60, 65, 70, 75, 81, 88, 92, 97, 103, 109, 117, 123, 130, 137, 144, 151, 158, 166, 175, 182, 189, 198, 207, 216, 226, 237, 245, 254, 263, 272, 282, 293, 303, 314

Offset: 0

Sascha Kurz and Peter Munn, Oct 19 2021

nonn

A circular region qualifies if (1) 3 (or more) grid points are incident on its circumference, or (2) it is an adjustment of a circular region, D, as defined in (1), so as to exclude only one or only consecutive grid points on the circumference of D. (Any such points on the circumference of D can be excluded by perturbing the center and radius of D by compatible but arbitrarily small amounts.)
The sequence definition is designed to help investigate the extent to which terms of A051602 can be equalled using only circular regions, while facilitating quicker calculation of terms. At the time of first submission, it is not clear to the authors that the qualification on the circular regions excludes any otherwise permissible configuration of points. In the absence of this knowledge, the qualification allows for the desired quicker calculation.
See A051602 for more information, references and links related to the general problem.

			For the following examples, we refer to _Hugo Pfoertner_'s pictorial catalog of circles passing through 3 or more grid points (see links section). Each illustration in the catalog is headed by the relevant terms of the sequences that give the squared radii of the circles, e.g. "A192493(5) = 25, A192494(5) = 16". The last line underneath each illustration gives the number of grid points in the circular region, e.g. "4+3=7" indicates 7 grid points total, of which 3 are on the circumference.
For n = 10, in the Pfoertner catalog we see the only circular region with 10 points corresponds to A192493(8). From the points in the illustration for A192493(8), 7 squares can be formed. This matches A051602(10) = 7, the maximal number of squares that can be formed from 10 points, so a(10) = 7.
For n = 20, in the Pfoertner catalog the only circular region with 20 points corresponds to A192493(29). From the points in the illustration for A192493(29), 31 squares can be formed. The circular region corresponding to A192493(24) has 21 points. From the points in the illustration for A192493(24) with any circumferential point excluded to leave 20 points, 32 squares can be formed. From a comprehensive search not detailed here, we ascertain that 32 is the most squares that can be formed from a 20 point configuration defined in the specified manner, so a(20) = 32.

Sascha Kurz, Table of n, a(n) for n = 0..100
Sascha Kurz, C++ program
Hugo Pfoertner, Circles Passing through 3 Points of the Square Lattice, illustrations up to R^2=10.

Cf. A051602, A192493/A192494.

a(n) <= A051602(n).

A135708 Minimal total number of edges in a polyhex consisting of n hexagonal cells.

Original entry on oeis.org

6, 11, 15, 19, 23, 27, 30, 34, 38, 41, 45, 48, 52, 55, 59, 62, 66, 69, 72, 76, 79, 83, 86, 89, 93, 96, 99, 103, 106, 109, 113, 116, 119, 123, 126, 129, 132, 136, 139, 142, 146, 149, 152, 155, 159, 162, 165, 168, 172, 175, 178, 181, 185, 188, 191, 194, 198, 201, 204, 207, 210

Offset: 1

N. J. A. Sloane, based on an email from Sascha Kurz, Mar 05 2008

nonn

The extremal examples were described by Y. S. Kupitz in 1991.

Y. S. Kupitz, "On the maximal number of appearances of the minimal distance among n points in the plane", in Intuitive geometry: Proceedings of the 3rd international conference held in Szeged, Hungary, 1991; Amsterdam: North-Holland: Colloq. Math. Soc. Janos Bolyai. 63, 217-244.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A135711.

[3*n+Ceiling(Sqrt(12*n-3)): n in [1..65]]; // Vincenzo Librandi, Oct 30 2016

Table[3*n + Ceiling[Sqrt[12*n - 3]], {n,1,25}] (* G. C. Greubel, Oct 29 2016 *)

a(n) = 3*n + ceil(sqrt(12*n-3)); \\ Michel Marcus, Oct 30 2016

from math import isqrt
def A135708(n): return 3*n+1+isqrt(12*n-4) # Chai Wah Wu, Jul 28 2022

a(n) = 3*n + ceiling(sqrt(12*n - 3)). - H. Harborth

2*a(n) - A135711(n) = 6n. - Tanya Khovanova, Mar 07 2008

A120103 Number of polyominoes consisting of 9 regular unit n-gons.

Original entry on oeis.org

160, 1285, 14445, 6572, 14982, 65323, 280014, 664411, 1908239, 1314914, 1968684, 4158216, 9707046, 17054708, 33522023, 26019735, 33942901, 56537856, 100952307, 153177526, 251530341, 208524646, 254079408, 374310135, 586169115, 812395658

Offset: 3

Sascha Kurz, Jun 09 2006

nonn

			a(3)=160 because there are 160 polyiamonds consisting of 9 triangles and a(4)=1285 because there are 1285 polyominoes consisting of 9 squares.

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, preprint, arXiv:math.CO/0605144

Cf. A000577, A000104, A000105, A000228, A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A103465, A120102, A120104.

A120102 Number of polyominoes consisting of 8 regular unit n-gons.

Original entry on oeis.org

66, 369, 2812, 1448, 2876, 10102, 34838, 73675, 181127, 131801, 185297, 352375, 725869, 1180526, 2104485, 1694978, 2123088, 3291481, 5402087, 7739008, 11832175, 10079003, 11917261, 16624712, 24389611, 32317393, 45260884

Offset: 3

Sascha Kurz, Jun 09 2006

nonn

			a(3)=66 because there are 66 polyiamonds consisting of 8 triangles and a(4)=369 because there are 369 polyominoes consisting of 8 squares.

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, preprint, arXiv:math/0605144 [math.CO], 2006.

Cf. A000577, A000104, A000228, A103465, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A103465, A120103, A120104.

A122133 Number of different polyominoes with maximum area of the convex hull.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 9, 26, 22, 53, 36, 93, 64, 151, 94, 228, 143, 329, 195, 455, 271, 611, 351, 798, 460, 1021, 574, 1281, 722, 1583, 876, 1928, 1069, 2321, 1269, 2763, 1513, 3259, 1765, 3810, 2066, 4421, 2376, 5093, 2740, 5831, 3114, 6636, 3547, 7513, 3991

Offset: 1

Sascha Kurz, Aug 21 2006

Colin Barker, Table of n, a(n) for n = 1..1000
K. Bezdek, P. Brass and H. Harborth, Maximum convex hulls of connected systems of segments and of polyominoes, Beiträge Algebra Geom., Vol. 35(1) (1994), pp. 37-43.
S. Kurz, Polyominoes with maximum convex hull, Diploma thesis, Bayreuth (2004).
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,-4,0,1,0,2,0,-1).

A122133 := proc(n)
    if modp(n,4)= 0 then
        (n^3-2*n^2+4*n)/16 ;
    elif modp(n,4)= 1 then
        (n^3-2*n^2+13*n+20)/32 ;
    elif modp(n,4)= 2 then
        (n^3-2*n^2+4*n+8)/16 ;
    else
        (n^3-2*n^2+5*n+8)/32 ;
    fi;
end proc: # R. J. Mathar, May 19 2019

Vec(x*(1+x-x^2+x^3+2*x^4+4*x^5+2*x^6+5*x^7+2*x^8+x^9)/((1-x)^4*(1+x)^4*(1+x^2)^2) + O(x^80)) \\ Colin Barker, Oct 14 2016

a(n) = (n^3 - 2*n^2 + 4*n)/16 if n mod 4 = 0;
a(n) = (n^3 - 2*n^2 + 13*n + 20)/32 if n mod 4 = 1;
a(n) = (n^3 - 2*n^2 + 4*n + 8)/16 if n mod 4 = 2;
a(n) = (n^3 - 2*n^2 + 5*n + 8)/32 if n mod 4 = 3.
G.f.: (1 + x - x^2 - x^3 + 2*x^5 + 8*x^6 + 2*x^7 + 4*x^8 + 2*x^9 - x^10 + x^12)/((1-x^2)^2*(1-x^4)^2).
From Luce ETIENNE, Aug 14 2019: (Start)
a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
a(n) = 2*a(n-2) + a(n-4) - 4*a(n-6) + a(n-8) + 2*a(n-10) - a(n-12).
a(n) = (3*n^3 - 6*n^2 + 17*n + 22 + (n^3 - 2*n^2 - n - 6)*(-1)^n - 4*(4*cos(n*Pi/2) - (2*n+3)*sin(n*Pi/2)))/64. (End)
E.g.f.: (1/64)*(-exp(-x)*(6 - 2*x - x^2 + x^3) + exp(x)*(22 + 14*x + 3*x^2 + 3*x^3) - 4*(4*cos(x) - 2*x*cos(x) - 3*sin(x))). - Stefano Spezia, Aug 14 2019

A120104 Number of polyominoes consisting of 10 regular unit n-gons.

Original entry on oeis.org

448, 4655, 76092, 30490, 80075, 430302, 2285047, 6078768, 20376032, 13303523, 21208739, 49734303, 131517548, 249598727, 540742895, 404616118, 549711709, 983715865, 1910489463, 3070327312

Offset: 3

Sascha Kurz, Jun 09 2006

nonn

			a(3)=448 because there are 448 polyiamonds consisting of 10 triangles;
a(4)=4655 because there are 4655 polyominoes consisting of 10 squares.

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, preprint arXiv:math/0605144 [math.CO], 2006.

Cf. A000577, A000104, A000105, A000228, A103465 - A103473, A103465, A120102, A120103.

A105659 Number of different characteristics, this is the squarefree part of (a+b+c)(a+b-c)(a-b+c)(-a+b+c), of integral triangles (a,b,c) with diameter n.

Original entry on oeis.org

1, 2, 4, 5, 8, 11, 12, 16, 18, 22, 28, 28, 35, 38, 49, 50, 57, 65, 75, 74, 87, 83, 112, 111, 114, 120, 135, 146, 175, 168, 196, 185, 213, 222, 219, 234, 267, 270, 293, 306, 339, 333, 386, 348, 365, 420, 460, 431, 445, 436, 490, 480, 577, 511, 549, 559, 610, 635

Offset: 1

Sascha Kurz, May 04 2005

nonn

			a(3)=4 because the integral triangles with diameter 3 are (3,2,2), (3,3,1), (3,3,2), (3,3,3) and the characteristics are 7, 35, 2, 3.

SquareFreePart[n_] := Times @@ Apply[Power, ({#1[[1]], Mod[#1[[2]], 2]} & ) /@ FactorInteger[n], {1}]; SquareFreePart[{a_, b_, c_}] := SquareFreePart[ (a+b+c)*(a+b-c)*(a-b+c)*(-a+b+c)]; ok[{a_, b_, c_}] := a-b < c < a+b && a-c < b < a+c && b-c < a < b+c; triangles[a_] := Reap[Do[ If[ok[{a, b, c}], Sow[{a, b, c}]], {b, 1, a}, {c, 1, b}]][[ 2, 1]]; a[n_] := Length[ Union[ SquareFreePart /@ triangles[n]]]; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Sep 11 2012 *)

A099122 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3}.

Original entry on oeis.org

1, 4, 55, 1540, 73815, 5461512, 581106988, 84431259000, 16104878212995, 3910294246315600, 1178924607035010836, 432472873725488656424, 189789513537655207705620, 98222259182333060014344720

Offset: 0

Sascha Kurz, Sep 28 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 3, where the referees cannot be distinguished.

a(n) is the number n element multisets of n element multisets of a 4-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=4 of A331436.

Cf. A099121, A099123, A099124, A099125, A099126, A099127, A099128.

a(n)={binomial(binomial(n+3, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n+3, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended by Andrew Howroyd, Jan 17 2020

A099123 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4}.

Original entry on oeis.org

1, 5, 120, 7770, 1088430, 286243776, 127860662755, 90079147136880, 94572327271677750, 141504997346476482290, 291098519807782284023426, 799388312264077003441393875, 2859142263297618955891805452700

Offset: 0

Sascha Kurz, Sep 28 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 4, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of a 5-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=5 of A331436.

Cf. A099121, A099122, A099124, A099125, A099126, A099127, A099128.

a(n)={binomial(binomial(n + 4, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n + 4, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended by Andrew Howroyd, Jan 17 2020

A099125 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4,5,6}.

Original entry on oeis.org

1, 7, 406, 102340, 83369265, 179224992408, 878487565272240, 8800321588119330984, 165564847349896309234920, 5470105884755875924791320090, 300550263698274781577833262263448, 26251679033395309424785182716562495776, 3509663406416043297299781592276029113718775

Offset: 0

Sascha Kurz, Sep 28 2004

nonn

This is the number of possible votes of n referees judging n dancers by a mark between 0 and 6, where the referees cannot be distinguished.

a(n) is the number of n element multisets of n element multisets of a 7-set. - Andrew Howroyd, Jan 17 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Column k=7 of A331436.

Cf. A099121, A099122, A099123, A099124, A099126, A099127, A099128.

a(n)={binomial(binomial(n + 6, n) + n - 1, n)} \\ Andrew Howroyd, Jan 17 2020

a(n) = binomial(binomial(n + 6, n) + n - 1, n). - Andrew Howroyd, Jan 17 2020

a(0)=1 prepended and a(11) and beyond from Andrew Howroyd, Jan 17 2020

cp's OEIS Frontend

User: Sascha Kurz

A348469 Maximal number of squares that can be formed from the grid points in a qualifying circular region of the plane that contains exactly n points of a square grid.

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A135708 Minimal total number of edges in a polyhex consisting of n hexagonal cells.

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A120103 Number of polyominoes consisting of 9 regular unit n-gons.

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A120102 Number of polyominoes consisting of 8 regular unit n-gons.

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A122133 Number of different polyominoes with maximum area of the convex hull.

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A120104 Number of polyominoes consisting of 10 regular unit n-gons.

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A105659 Number of different characteristics, this is the squarefree part of (a+b+c)(a+b-c)(a-b+c)(-a+b+c), of integral triangles (a,b,c) with diameter n.

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Mathematica

A099122 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3}.

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PARI

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A099123 Number of orbits of the wreath product of S_n with S_n on n X n matrices over {0,1,2,3,4}.

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