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 1-5 of 5 results.

A190098 T(n,k)=Number of rhombuses on a (n+1)X(k+1) grid.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 10, 10, 4, 5, 15, 22, 15, 5, 6, 20, 36, 36, 20, 6, 7, 26, 50, 66, 50, 26, 7, 8, 32, 66, 96, 96, 66, 32, 8, 9, 39, 82, 130, 151, 130, 82, 39, 9, 10, 46, 100, 164, 212, 212, 164, 100, 46, 10, 11, 54, 120, 204, 273, 312, 273, 204, 120, 54, 11, 12, 62, 142, 248, 344
Offset: 1

Views

Author

R. H. Hardin May 04 2011

Keywords

Comments

Table starts
..1..2...3...4...5...6....7....8....9...10...11...12...13...14...15...16...17
..2..6..10..15..20..26...32...39...46...54...62...71...80...90..100..111..122
..3.10..22..36..50..66...82..100..120..142..164..188..212..238..264..292..320
..4.15..36..66..96.130..164..204..248..296..344..396..448..504..560..620..680
..5.20..50..96.151.212..273..344..421..504..587..676..765..860..959.1064.1169
..6.26..66.130.212.312..412..527..650..782..914.1059.1204.1358.1520.1691.1862
..7.32..82.164.273.412..564..736..918.1112.1306.1520.1734.1960.2198.2448.2698
..8.39.100.204.344.527..736..984.1244.1520.1796.2104.2412.2736.3076.3438.3800
..9.46.120.248.421.650..918.1244.1601.1978.2355.2776.3197.3638.4103.4600.5097
.10.54.142.296.504.782.1112.1520.1978.2478.2978.3535.4092.4674.5288.5945.6602

Examples

			Some solutions for n=5 k=3
..3..1....1..2....2..3....0..2....3..1....4..2....1..1....0..1....1..1....1..1
..4..0....1..3....3..1....1..0....3..2....4..3....1..3....2..0....2..3....2..0
..5..1....2..3....5..0....3..1....4..2....5..3....3..3....4..1....4..2....3..1
..4..2....2..2....4..2....2..3....4..1....5..2....3..1....2..2....3..0....2..2

Links

Crossrefs

Diagonal is A189418(n+1)

Formula

Empirical for column k: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4) for n>4,11,11,27,27,51,51,83,83,123,123,171,171 respectively for k=2..14

A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

Original entry on oeis.org

0, 0, 4, 24, 84, 224, 516, 1068, 2016, 3528, 5832, 9256, 14208, 21180, 30728, 43488, 60192, 81660, 108828, 142764, 184708, 236088, 298476, 373652, 463524, 570228, 696012, 843312, 1014720, 1213096, 1441512, 1703352, 2002196, 2341848, 2726400, 3160272, 3648180
Offset: 0

Views

Author

Peter Kagey, May 06 2020

Keywords

Comments

a(n) >= 4 * A269747(n).
a(n) >= 4 * A000389(n+3) = A210569(n+2).
a(n) >= 4 * (n-1) + 4 * a(n-1) - 6 * a(n-2) + 4 * a(n-3) - a(n-4) for n >= 4.

Links

Crossrefs

Cf. A000292, A000389, A210569.
Cf. A000332 (equilateral triangles in triangular grid), A269747 (regular tetrahedra in a tetrahedral grid), A102698 (equilateral triangles in cube), A103158 (regular tetrahedra in cube).
Cf. A077435, A085582, A187452, A189412-A189418, A271910.

A334881 Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.

Original entry on oeis.org

0, 0, 6, 54, 240, 810, 2274, 5304, 10752, 19992, 34854, 57774, 91200, 139338, 206394, 296832, 417120, 575556, 779238, 1037514, 1359792, 1760694, 2251362, 2845140, 3554976, 4404876, 5416278, 6605946, 7996896, 9621678, 11500962, 13667772, 16143552, 18973608, 22190406
Offset: 0

Views

Author

Peter Kagey, May 14 2020

Keywords

Comments

a(n) >= 3*n*A002415(n).

Examples

			For n = 5, one such square has vertex set {(2,1,1), (5,4,1), (4,5,5), (1,2,5)}.

Links

Crossrefs

Cf. A002415 (squares in square grid), A098928 (cubes in cube grid).
Cf. A000332, A077435, A085582, A102698, A103158, A187452, A189412-A189418, A269747, A271910, A334581.

Extensions

a(7)-a(12) from Pontus von Brömssen, May 15 2020
a(13)-a(20) from Peter Kagey, Jul 29 2020 via Mathematics Stack Exchange link
Terms a(21) and beyond from Zachary Kaplan, Sep 01 2020, via Mathematics Stack Exchange link

A189415 Number of trapezoids on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 50, 490, 2618, 9519, 28432, 70796, 157912, 321161, 610482, 1082570, 1848362, 3003015, 4716792, 7204604, 10730528, 15530189, 22093410, 30723078, 42146178, 56981411, 75952240, 99685104, 129757248
Offset: 1

Views

Author

Martin Renner, Apr 21 2011

Keywords

Links

Crossrefs

Cf. A181945, A186434, A189413, A189414, A189416, A189418.

Extensions

a(6)-a(25) from Nathaniel Johnston, Apr 25 2011

A181947 Number of rhombi, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 3, 6, 11, 16, 24, 31, 43, 53, 67, 78, 99, 112, 132, 151, 179, 196, 226, 245, 282, 309, 341, 364, 416, 445, 483, 517, 570, 599, 659, 690, 754, 797, 847, 894, 975, 1012, 1068, 1119, 1211, 1252, 1338, 1381, 1466, 1536, 1604, 1651, 1775, 1833, 1923, 1990, 2091
Offset: 1

Views

Author

Martin Renner, Apr 03 2012

Keywords

Examples

			a(1) = 0 because the 1 X 1 grid has no rhombi.
a(2) = 1 because the 2 X 2 grid has one rhombus.
a(3) = 3 because the 3 X 3 grid has 3 congruence classes of rhombi (all of which are squares) out of 6 rhombi total.
a(3) = 6 because the 4 X 4 grid has 6 congruence classes of rhombi, out of 22 rhombi total:
+---------+---------+---------+
| . . . . | . . . . | . . . . |
| . . . . | o . o . | . o . . |
| o o . . | . . . . | o . o . |
| o o . . | o . o . | . o . . |
+---------+---------+---------+
| o . . o | . . . o | . o . . |
| . . . . | . o . . | . . . o |
| . . . . | . . o . | o . . . |
| o . . o | o . . . | . . o . |
+---------+---------+---------+

Links

Crossrefs

Cf. A181944, A189418.

Extensions

a(7)-a(53) from Lucas A. Brown, Feb 08 2024
Showing 1-5 of 5 results.

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