cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A331782 Total number of vertices in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

Original entry on oeis.org

3, 7, 21, 25, 63, 67, 129, 133, 219, 223, 333, 337, 471, 475, 621, 637, 819, 823, 1029, 1021, 1263, 1267, 1521, 1477, 1803, 1807, 2109, 2113, 2439, 2431, 2793, 2797, 3171, 3175, 3549, 3577, 3999, 4003, 4449, 4417, 4923, 4903, 5421, 5425, 5859, 5947, 6489, 6397, 7059, 7063, 7653, 7657, 8271, 8275, 8889
Offset: 1

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Author

N. J. A. Sloane, Feb 13 2020

Keywords

Comments

a(n) <= 3*(n^2-n+1), with equality iff n is odd and not a member of A332378. A331423 gives the difference between a(n) and the upper bound.

Links

Crossrefs

Cf. A091908, A092098 (number of cells), A332376 (number of edges), A332378, A331423.

Formula

a(n) = A091908(n) + 3*n.

A332378 Odd numbers k such that A091908(k) < 3*(k-1)^2.

Original entry on oeis.org

15, 35, 45, 55, 63, 65, 75, 77, 85, 91, 99, 105, 117, 119, 135, 143, 153, 161, 165, 171, 175, 187, 189, 195, 203, 207, 209, 221, 225, 231, 245, 247, 255, 259, 261, 273, 275, 285, 297, 299, 315, 319, 323, 325, 333, 341, 345, 351, 357, 369, 375, 377, 385, 387, 391
Offset: 1

Views

Author

N. J. A. Sloane, Feb 14 2020

Keywords

Comments

These are the odd numbers k such that A331423(k) != 0.
If k is even then always A091908(k) < 3*(k-1)^2.

Links

Crossrefs

Cf. A091908, A331423.

Extensions

More terms from Jinyuan Wang, Feb 17 2020

A331425 Divide each side of a triangle into 2*n (n>=1) equal parts and trace the corresponding cevians, i.e., join every point, except for the first and last ones, with the opposite vertex. a(n) is the number of points at which three cevians meet.

Original entry on oeis.org

1, 7, 13, 19, 25, 31, 37, 43, 49, 61, 61, 91, 73, 79, 91, 91, 97, 103, 109, 133, 133, 127, 133, 187, 145, 151, 157, 175, 169, 235, 181, 187, 205, 199, 229, 283, 217, 223, 235, 325, 241, 283, 253, 271, 331, 271, 277, 343, 289, 301, 301, 319, 313, 319, 349, 439
Offset: 1

Views

Author

César Eliud Lozada, Jan 16 2020

Keywords

Comments

A bisection of A331423.

Crossrefs

Cf. A331423, A331428.

Programs

  • Mathematica
    CevIntersections[n_] := Length[Solve[(n - i)*(n - j)*(n - k) - i*j*k == 0 && 0 < i < n &&  0 < j < n && 0 < k < n, {i, j, k}, Integers]];
Map[CevIntersections[#] &, Range[2,50,2]]

Formula

a(n) = A331423(2*n).

A331428 Divide each side of a triangle into 2*n-1 (n>=1) equal parts and trace the corresponding cevians, i.e., join every point, except for the first and last ones, with the opposite vertex. a(n) is the number of points at which three cevians meet.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 42, 0, 0, 0, 0, 12, 0, 0, 0, 48, 6, 0, 0, 0, 0, 6, 12, 0, 0, 0, 6, 0, 0, 12, 0, 0, 0, 24, 0, 0, 90, 0, 0, 0, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 24, 0, 0, 0, 0, 12, 0, 0, 0, 12, 0
Offset: 1

Views

Author

César Eliud Lozada, Jan 16 2020

Keywords

Comments

A bisection of A331423.

Crossrefs

Cf. A331423, A331425.

Programs

  • Mathematica
    CevIntersections[n_] := Length[Solve[(n - i)*(n - j)*(n - k) - i*j*k == 0 && 0 < i < n &&  0 < j < n && 0 < k < n, {i, j, k}, Integers]];
Map[CevIntersections[#] &, Range[1,51,2]]

Formula

a(n) = A331423(2*n-1).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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