Eckard Specht - cp's OEIS frontend

A063781 a(n) is the number of pairs of integer quadruples (b_1, b_2, b_3, b_4) and (c_1, c_2, c_3, c_4) satisfying 1 <= b_1 < b_2 < b_3 < b_4 < n, 1 <= c_1 < c_2 < c_3 < c_4 < n, b_i != c_j for all i,j = 1,2,3,4 and Product_{i=1..4} sin(2Pib_i/n) = Product_{i=1..4} sin(2Pic_i/n).

0, 1, 3, 4, 5, 9, 18, 17, 41, 29, 84, 45, 167, 66, 253, 93, 386, 126, 534, 166, 782, 214, 966, 270, 1380, 335, 1601, 410, 3053, 495, 2448, 591, 3135, 699, 3546, 819, 4785, 952, 4947, 1099, 8350, 1260, 6660, 1436, 8804, 1628, 8724, 1836, 10620, 2061, 11191

Offset: 8

Eckard Specht, Aug 17 2001

nonn

			For n=9, the only solution is (1, 4, 6, 7), (2, 3, 5, 8). - _Sean A. Irvine_, May 30 2023

Eckard Specht, Table of n, a(n) for n = 8..200

Cf. A063780.

Revised by Sean A. Irvine, May 30 2023

A063780 a(n) is the number of pairs of integer quadruples (b_1, b_2, b_3, b_4) and (c_1, c_2, c_3, c_4) satisfying 1 <= b_1 < b_2 < b_3 < b_4 < n, 1 <= c_1 < c_2 < c_3 < c_4 < n, b_i != c_j for all i,j = 1,2,3,4 and Product_{i=1..4} cos(2Pib_i/n) = Product_{i=1..4} cos(2Pic_i/n).

Original entry on oeis.org

0, 1, 3, 4, 9, 9, 18, 17, 93, 29, 84, 45, 433, 66, 253, 93, 1274, 126, 534, 166, 2940, 214, 1120, 270, 5866, 335, 1601, 410, 11359, 495, 2448, 591, 17371, 699, 3654, 819, 27487, 954, 4947, 1099, 42980, 1260, 6660, 1436, 59356, 1628, 8832, 1836, 82224, 2061

Offset: 8

Eckard Specht, Aug 17 2001

nonn

			For n=9, the only solution is (1, 4, 6, 7), (2, 3, 5, 8). - _Sean A. Irvine_, May 30 2023

Eckard Specht, Table of n, a(n) for n = 8..200
Sean A. Irvine, Java program (github)
Eckard Specht, C++ program

Cf. A063781.

Revised by Sean A. Irvine, May 30 2023

A051657 Experimental values for number of equal circles that are packed into a square for which the density of the packing is strictly increasing.

Original entry on oeis.org

1, 30, 38, 39, 52, 67, 68, 99, 119, 120

Offset: 1

Eckard Specht (eckard.specht(AT)physik.uni-magdeburg.de)

nonn

H. T. Croft, K. J. Falconer and R. K. Guy: Unsolved problems in geometry, Springer, New York, 1991.

D. Boll, Optimal Packing Of Circles And Spheres.
E. Friedman, Erich's Packing Center.
C. D. Maranas, C. A. Floudas and P. M. Pardalos, New results in the packing of equal circles in a square, Discrete Mathematics 142 (1995), p. 287-293.
K. J. Nurmela and Patric R. J. Östergård, Packing up to 50 equal circles in a square, Discrete Comput. Geom. 18 (1997) 1, p. 111-120.
E. Specht, www.packomania.com.

I do not know how many of these values have been rigorously proved. - N. J. A. Sloane

A051660 Experimental values for number of circles in packing equal circles into a square for which there is a loose circle.

Original entry on oeis.org

7, 11, 13, 14, 17, 19, 21, 22, 26, 28, 29, 31, 32, 33, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 85, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100

Offset: 0

Eckard Specht (eckard.specht(AT)physik.uni-magdeburg.de)

nonn

H. T. Croft, K. J. Falconer and R. K. Guy: Unsolved problems in geometry, Springer, New York, 1991.

D. Boll, Optimal Packing Of Circles And Spheres
L. G. Casado, I. García, P. G. Szabó, and T. Csendes, Packing Equal Circles in a Square II. - New Results for up to 100 Circles Using the TAMSASS-PECS Algorithm, Optimization Theory: Recent Developments from Mátraháza, Kluwer Academic Publishers, Dordrecht, 2001, pp. 207-224.
E. Friedman, Erich's Packing Center
C. D. Maranas, C. A. Floudas and P. M. Pardalos, New results in the packing of equal circles in a square, Discrete Mathematics 142 (1995), p. 287-293.
K. J. Nurmela and Patric R. J. Östergård, Packing up to 50 equal circles in a square, Discrete Comput. Geom. 18 (1997) 1, p. 111-120.
E. Specht, www.packomania.com
P. G. Szabó, Packing up to 100 circles in a square.
P. G. Szabó, T. Csendes, L. G. Casado, and I. García, Packing Equal Circles in a Square I. - Problem Setting and Bounds for Optimal Solutions, Optimization Theory: Recent Developments from Mátraháza, Kluwer Academic Publishers, Dordrecht, 2001, pp. 191-206.

I do not know how many of these values have been rigorously proved. - N. J. A. Sloane

A051658 Experimental values for maximal number of contacts between equal circles and the box that are packed into a square.

Original entry on oeis.org

4, 5, 7, 12, 12, 13, 14, 20, 24, 21, 20, 25, 25, 32, 36, 40, 34, 38, 37, 44, 39, 43, 56, 56, 60, 56, 55, 57, 65, 65, 55, 63, 65, 80, 80, 84, 77, 77, 80, 85, 100, 90, 85, 82, 94, 91, 94, 111, 120, 100, 97, 105, 110, 115, 113, 119, 113

Offset: 1

Eckard Specht (eckard.specht(AT)physik.uni-magdeburg.de)

nonn

H. T. Croft, K. J. Falconer and R. K. Guy: Unsolved problems in geometry, Springer, New York, 1991.

D. Boll, Optimal Packing Of Circles And Spheres
L. G. Casado, I. García, P. G. Szabó, and T. Csendes, Packing Equal Circles in a Square II. - New Results for up to 100 Circles Using the TAMSASS-PECS Algorithm, Optimization Theory: Recent Developments from Mátraháza, Kluwer Academic Publishers, Dordrecht, 2001, pp. 207-224.
E. Friedman, Erich's Packing Center
C. D. Maranas, C. A. Floudas and P. M. Pardalos, New results in the packing of equal circles in a square, Discrete Mathematics 142 (1995), p. 287-293.
K. J. Nurmela and Patric R. J. Östergård, Packing up to 50 equal circles in a square, Discrete Comput. Geom. 18 (1997) 1, p. 111-120.
E. Specht, www.packomania.com
P. G. Szabó, Packing up to 100 circles in a square.
P. G. Szabó, T. Csendes, L. G. Casado, and I. García, Packing Equal Circles in a Square I. - Problem Setting and Bounds for Optimal Solutions, Optimization Theory: Recent Developments from Mátraháza, Kluwer Academic Publishers, Dordrecht, 2001, pp. 191-206.

I do not know how many of these values have been rigorously proved. - N. J. A. Sloane

A051659 Experimental values for maximal number of "loose" circles in packing equal circles into a square.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 1, 0, 0, 1, 0, 2, 0, 2, 1, 0, 0, 0, 2, 0, 1, 1, 0, 4, 3, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 1, 4, 3, 1

Offset: 1

Eckard Specht (eckard.specht(AT)physik.uni-magdeburg.de)

nonn

H. T. Croft, K. J. Falconer and R. K. Guy: Unsolved problems in geometry, Springer, New York, 1991.

D. Boll, Optimal Packing Of Circles And Spheres
E. Friedman, Erich's Packing Center
C. D. Maranas, C. A. Floudas and P. M. Pardalos, New results in the packing of equal circles in a square, Discrete Mathematics 142 (1995), p. 287-293.
K. J. Nurmela and Patric R. J. Östergård, Packing up to 50 equal circles in a square, Discrete Comput. Geom. 18 (1997) 1, p. 111-120.
E. Specht, www.packomania.com

I do not know how many of these values have been rigorously proved. - N. J. A. Sloane

A051661 Experimental values for number of circles in packing equal circles into a square for which there are no loose circles.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 23, 24, 25, 27, 30, 34, 35, 36, 37, 38, 39, 42, 52, 56, 67, 68, 77, 80, 86, 87, 99, 120, 137, 143, 150, 188

Offset: 0

Eckard Specht (eckard.specht(AT)physik.uni-magdeburg.de)

nonn

H. T. Croft, K. J. Falconer and R. K. Guy: Unsolved problems in geometry, Springer, New York, 1991.

D. Boll, Optimal Packing Of Circles And Spheres
L. G. Casado, I. García, P. G. Szabó, and T. Csendes, Packing Equal Circles in a Square II. - New Results for up to 100 Circles Using the TAMSASS-PECS Algorithm, Optimization Theory: Recent Developments from Mátraháza, Kluwer Academic Publishers, Dordrecht, 2001, pp. 207-224.
E. Friedman, Erich's Packing Center
C. D. Maranas, C. A. Floudas and P. M. Pardalos, New results in the packing of equal circles in a square, Discrete Mathematics 142 (1995), p. 287-293.
K. J. Nurmela and Patric R. J. Östergård, Packing up to 50 equal circles in a square, Discrete Comput. Geom. 18 (1997) 1, p. 111-120.
E. Specht, www.packomania.com
P. G. Szabó, Packing up to 100 circles in a square.
P. G. Szabó, T. Csendes, L. G. Casado, and I. García, Packing Equal Circles in a Square I. - Problem Setting and Bounds for Optimal Solutions, Optimization Theory: Recent Developments from Mátraháza, Kluwer Academic Publishers, Dordrecht, 2001, pp. 191-206.

Complement of A051660.

I do not know how many of these values have been rigorously proved. - N. J. A. Sloane

cp's OEIS Frontend

User: Eckard Specht

A063781 a(n) is the number of pairs of integer quadruples (b_1, b_2, b_3, b_4) and (c_1, c_2, c_3, c_4) satisfying 1 <= b_1 < b_2 < b_3 < b_4 < n, 1 <= c_1 < c_2 < c_3 < c_4 < n, b_i != c_j for all i,j = 1,2,3,4 and Product_{i=1..4} sin(2*Pi*b_i/n) = Product_{i=1..4} sin(2*Pi*c_i/n).

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A063780 a(n) is the number of pairs of integer quadruples (b_1, b_2, b_3, b_4) and (c_1, c_2, c_3, c_4) satisfying 1 <= b_1 < b_2 < b_3 < b_4 < n, 1 <= c_1 < c_2 < c_3 < c_4 < n, b_i != c_j for all i,j = 1,2,3,4 and Product_{i=1..4} cos(2*Pi*b_i/n) = Product_{i=1..4} cos(2*Pi*c_i/n).

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A051657 Experimental values for number of equal circles that are packed into a square for which the density of the packing is strictly increasing.

Views

Author

Keywords

References

Links

Extensions

A051660 Experimental values for number of circles in packing equal circles into a square for which there is a loose circle.

Views

Author

Keywords

References

Links

Extensions

A051658 Experimental values for maximal number of contacts between equal circles and the box that are packed into a square.

Views

Author

Keywords

References

Links

Extensions

A051659 Experimental values for maximal number of "loose" circles in packing equal circles into a square.

Views

Author

Keywords

References

Links

Extensions

A051661 Experimental values for number of circles in packing equal circles into a square for which there are no loose circles.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A063781 a(n) is the number of pairs of integer quadruples (b_1, b_2, b_3, b_4) and (c_1, c_2, c_3, c_4) satisfying 1 <= b_1 < b_2 < b_3 < b_4 < n, 1 <= c_1 < c_2 < c_3 < c_4 < n, b_i != c_j for all i,j = 1,2,3,4 and Product_{i=1..4} sin(2Pib_i/n) = Product_{i=1..4} sin(2Pic_i/n).

A063780 a(n) is the number of pairs of integer quadruples (b_1, b_2, b_3, b_4) and (c_1, c_2, c_3, c_4) satisfying 1 <= b_1 < b_2 < b_3 < b_4 < n, 1 <= c_1 < c_2 < c_3 < c_4 < n, b_i != c_j for all i,j = 1,2,3,4 and Product_{i=1..4} cos(2Pib_i/n) = Product_{i=1..4} cos(2Pic_i/n).