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.

A028419 Congruence classes of triangles which can be drawn using lattice points in n X n grid as vertices.

Original entry on oeis.org

0, 1, 8, 29, 79, 172, 333, 587, 963, 1494, 2228, 3195, 4455, 6050, 8032, 10481, 13464, 17014, 21235, 26190, 31980, 38666, 46388, 55144, 65131, 76449, 89132, 103337, 119184, 136757, 156280, 177796, 201430, 227331, 255668, 286606, 320294, 356884, 396376, 439100, 485427, 535049, 588457, 645803
Offset: 0

Views

Author

David J. Rusin

Keywords

Links

Crossrefs

Cf. A028492, A189979, A190021, A190022, A189978, A190313, A189978, A190313.

Programs

  • Maple
    a:=proc(n) local TriangleSet,i,j,k,l,A,B,C; TriangleSet:={}: for i from 0 to n do for j from 0 to n do for k from 0 to n do for l from 0 to n do A:=i^2+j^2: B:=k^2+l^2: C:=(i-k)^2+(j-l)^2: if A^2+B^2+C^2<>2*(A*B+B*C+C*A) then TriangleSet:={op(TriangleSet),sort([sqrt(A),sqrt(B),sqrt(C)])}: fi: od: od: od: od: return(nops(TriangleSet)); end: # Martin Renner, May 03 2011

Extensions

More terms from Chris Cole (chris(AT)questrel.com), Jun 28 2003
a(36)-a(39) from Martin Renner, May 08 2011

A190021 Number of acute triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 2, 8, 23, 51, 101, 179, 295, 460, 688, 988, 1382, 1876, 2495, 3258, 4191, 5298, 6613, 8166, 9973, 12065, 14472, 17208, 20341, 23873, 27838, 32282, 37238, 42734, 48840, 55573, 62973, 71067, 79934, 89640, 100172, 111613, 123959, 137336, 151842
Offset: 1

Views

Author

Martin Renner, May 04 2011

Keywords

Examples

			For n = 3 the two acute triangles are:
*..   .*.
..*   *..
*..   ..*

Links

Crossrefs

Cf. A028419, A189979, A190022.

Programs

  • Maple
    Triangles:=proc(n) local TriangleSet, i, j, k, l, A, B, C; TriangleSet:={}: for i from 0 to n do for j from 0 to n do for k from 0 to n do for l from 0 to n do A:=i^2+j^2: B:=k^2+l^2: C:=(i-k)^2+(j-l)^2: if A^2+B^2+C^2<>2*(A*B+B*C+C*A) then TriangleSet:={op(TriangleSet), sort([sqrt(A), sqrt(B), sqrt(C)])}: fi: od: od: od: od: return(TriangleSet); end:
IsAcuteTriangle:=proc(T) if T[1]^2+T[2]^2>T[3]^2 and T[1]^2+T[3]^2>T[2]^2 and T[2]^2+T[3]^2>T[1]^2 then true else false fi: end:
a:=proc(n) local TriangleSet,AcuteTriangleSet,i; TriangleSet:=Triangles(n): AcuteTriangleSet:={}: for i from 1 to nops(TriangleSet) do if IsAcuteTriangle(TriangleSet[i]) then AcuteTriangleSet:={op(AcuteTriangleSet),TriangleSet[i]} fi: od: return(nops(AcuteTriangleSet)); end:

Formula

a(n) = A028419(n) - A189979(n) - A190022(n).

Extensions

a(21)-a(40) from Martin Renner, May 08 2011

A241234 Number of obtuse triangles, distinct up to congruence, on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 1, 15, 72, 220, 528, 1076, 1965, 3314, 5254, 7954, 11589, 16306, 22349, 29939, 39305, 50738, 64462, 80775, 100096, 122602, 148748, 178834, 213268, 252439, 296857, 346878, 402778, 465346, 534966, 611966, 697143, 790749, 893638, 1006128, 1129134, 1262932, 1408116, 1565674
Offset: 1

Views

Author

Martin Renner, Apr 17 2014

Keywords

Comments

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

Examples

			For n = 2 the only kind of non-congruent obtuse triangles is the following:
/* *
. . *
\. .

Links

Crossrefs

Cf. A190022, A241226.

Formula

a(n) = A241231(n) - A241232(n) - A241233(n)

Extensions

a(7) from Martin Renner, May 31 2014
a(8)-a(14) from Giovanni Resta, May 31 2014
More terms from Bert Dobbelaere, Oct 17 2022

A190310 Number of obtuse isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 9, 12, 19, 24, 32, 37, 51, 57, 69, 80, 99, 107, 127, 136, 161, 176, 196, 207, 246, 262, 286, 306, 343, 357, 399, 414, 460, 485, 517, 544, 605, 623, 659, 689, 757
Offset: 1

Views

Author

Martin Renner, May 08 2011

Keywords

Links

Crossrefs

Cf. A028419, A108279, A189978, A190022, A190309.

Formula

a(n) = A189978(n) - A190309(n) - A108279(n).
Showing 1-4 of 4 results.

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