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 11-16 of 16 results.

A271915 Number of ways to choose three distinct points from a 5 X n grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, 2188, 2548, 2936, 3332, 3756, 4192, 4656, 5128, 5628, 6136, 6672, 7216, 7788, 8368, 8976, 9592, 10236, 10888, 11568, 12256, 12972, 13696, 14448, 15208, 15996, 16792
Offset: 1

Views

Author

N. J. A. Sloane, Apr 24 2016

Keywords

Links

Crossrefs

Row 5 of A271910.
Cf. A186434, A187452.

Programs

  • Mathematica
    Join[{0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, 2188, 2548, 2936, 3332}, LinearRecurrence[{2, 0, -2, 1}, {3756, 4192, 4656, 5128}, 20]] (* Jean-François Alcover, Sep 03 2018 *)

Formula

Conjectured g.f.: 4*x* (x^16-x^14+2*x^10+2*x^9-x^8-x^7 + 5*x^6+6*x^5+6*x^4+x^3-8*x^2-15*x-6) /((x+1)*(x-1)^3).
Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 18.
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016

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

A279414 a(n) is the total number of isosceles triangles having a bounding box n X k where k is in the range 1 to n inclusive.

Original entry on oeis.org

0, 4, 14, 22, 36, 48, 58, 66, 104, 100, 110, 118, 164, 148, 174, 174, 232, 200, 266, 226, 300, 272, 290, 282, 412, 332, 362, 358, 440, 376, 494, 386, 572, 464, 490, 490, 660, 476, 546, 562, 756, 552, 718, 582, 760, 692, 682, 634, 1004, 716, 862, 746, 900, 744
Offset: 1

Views

Author

Lars Blomberg, Feb 16 2017

Keywords

Comments

Row sums of A279413.

Links

Crossrefs

Cf. A186434, A187452, A271910-A271913, A271915, A279413.

A189895 T(n,k) = Number of isosceles right triangles on a (n+1) X (k+1) grid.

Original entry on oeis.org

4, 10, 10, 16, 28, 16, 22, 50, 50, 22, 28, 74, 96, 74, 28, 34, 98, 150, 150, 98, 34, 40, 122, 208, 244, 208, 122, 40, 46, 146, 268, 350, 350, 268, 146, 46, 52, 170, 328, 464, 516, 464, 328, 170, 52, 58, 194, 388, 582, 700, 700, 582, 388, 194, 58, 64, 218, 448, 702, 896, 968
Offset: 1

Views

Author

R. H. Hardin, Apr 30 2011

Keywords

Comments

Table starts
..4..10..16..22...28...34...40...46...52...58...64...70...76....82....88....94
.10..28..50..74...98..122..146..170..194..218..242..266..290...314...338...362
.16..50..96.150..208..268..328..388..448..508..568..628..688...748...808...868
.22..74.150.244..350..464..582..702..822..942.1062.1182.1302..1422..1542..1662
.28..98.208.350..516..700..896.1100.1308.1518.1728.1938.2148..2358..2568..2778
.34.122.268.464..700..968.1260.1570.1892.2222.2556.2892.3228..3564..3900..4236
.40.146.328.582..896.1260.1664.2100.2560.3038.3528.4026.4528..5032..5536..6040
.46.170.388.702.1100.1570.2100.2680.3300.3952.4628.5322.6028..6742..7460..8180
.52.194.448.822.1308.1892.2560.3300.4100.4950.5840.6762.7708..8672..9648.10632
.58.218.508.942.1518.2222.3038.3952.4950.6020.7150.8330.9550.10802.12078.13372

Examples

			Some solutions for n=7 k=5
..3..5....1..1....5..4....6..4....5..1....4..4....3..2....2..5....4..3....2..3
..1..4....2..4....1..3....3..5....4..3....1..1....4..1....0..1....2..3....0..0
..4..3....4..0....6..0....5..1....7..2....7..1....4..3....6..3....4..1....5..1

Links

Crossrefs

Diagonal is A187452(n+1).
(2n-1,n) diagonal is A189894.

Formula

Empirical for column k: a(n) = k*(k+1)*(k+2)*n + b(k) for n>2*k-2.
k=1: a(n) = 6*n - 2
k=2: a(n) = 24*n - 22 for n>2
k=3: a(n) = 60*n - 92 for n>4
k=4: a(n) = 120*n - 258 for n>6
k=5: a(n) = 210*n - 582 for n>8
k=6: a(n) = 336*n - 1140 for n>10
k=7: a(n) = 504*n - 2024 for n>12
k=8: a(n) = 720*n - 3340 for n>14
k=9: a(n) = 990*n - 5210 for n>16
k=10: a(n) = 1320*n - 7770 for n>18
k=11: a(n) = 1716*n - 11172 for n>20
k=12: a(n) = 2184*n - 15582 for n>22
k=13: a(n) = 2730*n - 21182 for n>24
k=14: a(n) = 3360*n - 28168 for n>26

A271911 Number of ways to choose three distinct points from a 2 X n grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 4, 10, 16, 24, 32, 42, 52, 64, 76, 90, 104, 120, 136, 154, 172, 192, 212, 234, 256, 280, 304, 330, 356, 384, 412, 442, 472, 504, 536, 570, 604, 640, 676, 714, 752, 792, 832, 874, 916, 960, 1004, 1050, 1096, 1144, 1192, 1242, 1292, 1344, 1396, 1450, 1504
Offset: 1

Views

Author

N. J. A. Sloane, Apr 24 2016

Keywords

Examples

			n=3: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, so a(3) = 10.

Links

Crossrefs

Row 2 of A271910.
Cf. A186434, A187452.
Same start as, but totally different from, 2*A213707.

Programs

  • Mathematica
    LinearRecurrence[{2,0,-2,1},{0,4,10,16},60] (* Harvey P. Dale, May 10 2018 *)

Formula

Conjectured g.f.: 2*x*(2*x^2-x-2)/((x+1)*(x-1)^3). It would be nice to have a proof!
Conjectures from Colin Barker, Apr 24 2016: (Start)
a(n) = (-1+(-1)^n+16*n+2*n^2)/4, or equivalently, a(n) = (n^2+8*n)/2 if n even, (n^2+8*n-1)/2 if n odd.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>4. (End)
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016
a(n) = round(n*(n/2+3)) - 4. - Bill McEachen, Aug 10 2025

Extensions

More terms from Harvey P. Dale, May 10 2018

A271912 Number of ways to choose three distinct points from a 3 X n grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 10, 36, 68, 108, 150, 200, 252, 312, 374, 444, 516, 596, 678, 768, 860, 960, 1062, 1172, 1284, 1404, 1526, 1656, 1788, 1928, 2070, 2220, 2372, 2532, 2694, 2864, 3036, 3216, 3398, 3588, 3780, 3980, 4182, 4392, 4604, 4824, 5046, 5276, 5508, 5748, 5990, 6240, 6492, 6752, 7014, 7284, 7556
Offset: 1

Views

Author

N. J. A. Sloane, Apr 24 2016

Keywords

Examples

			n=2: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, so a(2) = 10.

Links

Crossrefs

Row 3 of A271910.
Cf. A186434, A187452.

Programs

  • Mathematica
    Join[{0, 10}, LinearRecurrence[{2, 0, -2, 1}, {36, 68, 108, 150}, 50]] (* Jean-François Alcover, Oct 10 2018 *)

Formula

Conjectured g.f.: 2*x*(2*x^4+4*x^3+2*x^2-8*x-5)/((x+1)*(x-1)^3).
Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 6.
Conjectures from Colin Barker, Apr 25 2016: (Start)
a(n) = (-3*(47+(-1)^n)+64*n+10*n^2)/4 for n>2.
a(n) = (5*n^2+32*n-72)/2 for n>2 and even.
a(n) = (5*n^2+32*n-69)/2 for n>2 and odd.
(End)
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016

Extensions

More terms from Jean-François Alcover, Oct 10 2018
Previous Showing 11-16 of 16 results.

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

Content is available under The OEIS End-User License Agreement.