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.

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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.

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

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