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.

Previous Showing 11-17 of 17 results.

A362233 Number of vertices among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes when each pair of points is connected by a circle and where the points lie at the ends of the circles' diameter.

Original entry on oeis.org

17, 249, 1381, 4745, 12581, 26861, 51649, 89357, 145501, 225621, 335497
Offset: 1

Views

Author

Scott R. Shannon, Apr 12 2023

Keywords

Comments

A circle is constructed for every pair of the 1 + 4n points, the two points lying at the ends of a diameter of the circle. The number of distinct circles constructed from the points is A139275(n).
No formula for a(n) is currently known.

Links

Crossrefs

Cf. A362234 (regions), A362235 (edges), A362236 (k-gons), A139275 (distinct circles), A354605, A359932.

Formula

a(n) = A362235(n) - A362234(n) + 1 by Euler's formula.

A362234 Number of regions among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes when each pair of points is connected by a circle and where the points lie at the ends of the circles' diameter.

Original entry on oeis.org

32, 372, 1804, 5772, 14660, 30816, 58232, 100080, 161700, 249200, 368384
Offset: 1

Views

Author

Scott R. Shannon, Apr 13 2023

Keywords

Comments

A circle is constructed for every pair of the 1 + 4n points, the two points lying at the ends of a diameter of the circle. The number of distinct circles constructed from the points is A139275(n).
No formula for a(n) is currently known.

Links

Crossrefs

Cf. A362233 (vertices), A362235 (edges), A362236 (k-gons), A139275 (distinct circles), A353782, A359933.

Formula

a(n) = A362235(n) - A362233(n) + 1 by Euler's formula.

A362235 Number of edges among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes when each pair of points is connected by a circle and where the points lie at the ends of the circles' diameter.

Original entry on oeis.org

48, 620, 3184, 10516, 27240, 57676, 109880, 189436, 307200, 474820, 703880
Offset: 1

Views

Author

Scott R. Shannon, Apr 13 2023

Keywords

Comments

A circle is constructed for every pair of the 1 + 4n points, the two points lying at the ends of a diameter of the circle. The number of distinct circles constructed from the points is A139275(n).
No formula for a(n) is currently known.
See A362233 and A362234 for images of the circles.

Crossrefs

Cf. A362233 (vertices), A362234 (regions), A362236 (k-gons), A139275 (distinct circles), A356358, A359934.

Formula

a(n) = A362234(n) + A362233(n) - 1 by Euler's formula.

A362236 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes when each pair of points is connected by a circle and where the points lie at the ends of the circles' diameter.

Original entry on oeis.org

12, 12, 8, 32, 204, 120, 16, 56, 928, 652, 156, 4, 8, 72, 2724, 2332, 504, 120, 16, 4, 96, 6416, 6120, 1648, 352, 20, 8, 128, 13356, 12444, 4156, 668, 52, 12, 208, 24348, 23892, 8148, 1488, 124, 24, 248, 41268, 41528, 14108, 2616, 276, 36, 336, 65684, 67272, 23372, 4592, 388, 52, 0, 4
Offset: 1

Views

Author

Scott R. Shannon, Apr 13 2023

Keywords

Comments

A circle is constructed for every pair of the 1 + 4n points, the two points lying at the ends of a diameter of the circle. The number of distinct circles constructed from the points is A139275(n).
See A362233 and A362234 for images of the circles.

Examples

			The table begins:
 12, 12, 8;
 32, 204, 120, 16;
 56, 928, 652, 156, 4, 8;
 72, 2724, 2332, 504, 120, 16, 4;
 96, 6416, 6120, 1648, 352, 20, 8;
 128, 13356, 12444, 4156, 668, 52, 12;
 208, 24348, 23892, 8148, 1488, 124, 24;
 248, 41268, 41528, 14108, 2616, 276, 36;
 336, 65684, 67272, 23372, 4592, 388, 52, 0, 4;
 384, 99440, 105260, 36028, 7316, 708, 60, 4;
 464, 144684, 156976, 54136, 10792, 1224, 100, 8;
 .
 .

Crossrefs

Cf. A362233 (vertices), A362234 (regions), A362235 (edges), A139275 (distinct circles), A361623, A359935.

Formula

Sum of row n = A362234(n).

A340171 List of X-coordinates of point moving along one of the arms of a counterclockwise double square spiral; A340172 gives Y-coordinates.

Original entry on oeis.org

0, 1, 1, 0, -1, -2, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -6, -6, -6, -6, -6, -6, -6, -6, -6, -6, -6, -5
Offset: 0

Views

Author

Rémy Sigrist, Dec 30 2020

Keywords

Comments

The odd function f such that f(n) = (a(n), A340172(n)) for any n >= 0 will visit exactly once every lattice point (so it is a bijection from Z to Z^2).

Examples

			The spiral starts as follows:
      +-----+-----+-----+-----+-----+
      .                             |
      .                             |
      .     +-----+-----+-----+     +
      .     |5     4     3    |2    |
      .     |                 |     |
            +     +-----+-----+     +
            |6    |      0     1    |     .
            |     |                 |     .
            +     +-----+-----+-----+     .
            |7                            .
            |                             .
            +-----+-----+-----+-----+-----+
             8     9     10    11    12    13
- so a(0) = a(3) = a(10) = 0,
-    a(1) = a(2) = a(11) = 1.

Links

Crossrefs

Cf. A000217, A014105, A046092, A001105, A139274, A139275, A174344, A340172.

Programs

  • PARI
    See Links section.

Formula

abs(a(n+1)-a(n)) + abs(A340172(n+1)-A340172(n)) = 1.
a(n) = A340172(n) iff n belongs to A001105.
a(n) = -A340172(n) iff n belongs to A046092.
a(n) = 2*A340172(n) iff n belongs to A139274.
2*a(n) = A340172(n) iff n belongs to A139275.
a(n) * A340172(n) = 0 iff n belongs to A000217.
a(n) = 0 iff n belongs to A014105.

A340172 List of Y-coordinates of point moving along one of the arms of a counterclockwise double square spiral; A340171 gives X-coordinates.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 0, -1, -2, -2, -2, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -6, -6, -6
Offset: 0

Views

Author

Rémy Sigrist, Dec 30 2020

Keywords

Comments

The odd function f such that f(n) = (A340171(n), a(n)) for any n >= 0 will visit exactly once every lattice point (so it is a bijection from Z to Z^2).

Examples

			The spiral starts as follows:
      +-----+-----+-----+-----+-----+
      .                             |
      .                             |
      .     +-----+-----+-----+     +
      .     |5     4     3    |2    |
      .     |                 |     |
            +     +-----+-----+     +
            |6    |      0     1    |     .
            |     |                 |     .
            +     +-----+-----+-----+     .
            |7                            .
            |                             .
            +-----+-----+-----+-----+-----+
             8     9     10    11    12    13
- so a(0) = a(1) = a(6) = 0,
-    a(2) = a(3) = a(4) = a(5) = 1.

Links

Crossrefs

Cf. A000217, A000384, A001105, A046092, A139274, A139275, A274923, A340171.

Programs

  • PARI
    See Links section.

Formula

abs(a(n+1)-a(n)) + abs(A340171(n+1)-A340171(n)) = 1.
a(n) = A340171(n) iff n belongs to A001105.
a(n) = - A340171(n) iff n belongs to A046092.
2*a(n) = A340171(n) iff n belongs to A139274.
a(n) = 2*A340171(n) iff n belongs to A139275.
a(n) * A340171(n) = 0 iff n belongs to A000217.
a(n) = 0 iff n belongs to A000384.

A257144 Numbers n not of the form x+y*x^2 for x>1 and y>0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 37, 40, 41, 43, 44, 45, 47, 49, 51, 53, 59, 60, 61, 63, 64, 65, 67, 69, 71, 73, 76, 77, 79, 81, 83, 85, 87, 88, 89, 91, 92, 95, 96, 97, 99, 101, 103, 104, 107, 108, 109, 112, 113, 115, 117, 119, 121
Offset: 1

Views

Author

Gionata Neri, Apr 16 2015

Keywords

Comments

Number n such that (d*k+1) /= (n/d), for k>0 and each value of d, where d is a divisor >1 of n.

Crossrefs

For numbers of the form x+y*x^2 with 0A002378 (y=1), A014105 (y=2), A049451 (y=3), A007742 (y=4), A202803 (y=5), A049453 (y=6), A092277 (y=7), A139275 (y=8), A154517 (y=9), A055437 (y=10). - Danny Rorabaugh, Apr 20 2015

Programs

  • Mathematica
    n = 71; Take[Complement[Range[n^2], DeleteDuplicates@ Sort@ Flatten@ Table[x + y x^2, {x, 2, n}, {y, 1, n}]], n] (* Michael De Vlieger, Apr 17 2015 *)
Previous Showing 11-17 of 17 results.

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