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.

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

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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

A190022 Number of obtuse triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 2, 12, 39, 95, 193, 355, 597, 943, 1426, 2071, 2904, 3977, 5306, 6956, 8963, 11370, 14225, 17587, 21515, 26053, 31310, 37282, 44061, 51785, 60436, 70127, 80939, 92952, 106267, 120982, 137124, 154841, 174225, 195366, 218394, 243457, 270505, 299749, 331441
Offset: 1

Views

Author

Martin Renner, May 04 2011

Keywords

Examples

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

Links

Crossrefs

Cf. A028419, A189979, A190021.
Cf. A103428.

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:
IsObtuseTriangle:=proc(T) if T[1]^2+T[2]^2

Formula

a(n) = A028419(n) - A190021(n) - A189979(n).

Extensions

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

A190180 Continued fraction of (1+sqrt(-3+4*sqrt(2)))/2.

Original entry on oeis.org

1, 3, 5, 1, 2, 1, 1, 1, 2, 1, 12, 1, 5, 1, 1, 2, 1, 14, 2, 9, 11, 1, 12, 1, 2, 1, 832, 1, 2, 2, 5, 1, 1, 17, 1, 2, 1, 9, 1, 12, 1, 1, 1, 6, 3, 2, 1, 1, 6, 3, 1, 1, 1, 2, 2, 1, 3, 1, 3, 3, 1, 2, 1, 45, 1, 1, 1, 1, 62, 9, 1, 1, 2, 3, 1, 6, 1, 3, 5, 1
Offset: 1

Views

Author

Clark Kimberling, May 05 2011

Keywords

Comments

Equivalent to the periodic continued fraction [1,r,1,r,...] where r=1+sqrt(2), the silver ratio. For geometric interpretations of both continued fractions, see A189979 and A188635.
1 followed by A190178.

Links

Crossrefs

Cf. A190179, A188635, A190177, A190178.

Programs

  • Magma
    ContinuedFraction((1+Sqrt(-3+4*Sqrt(2)))/2); // G. C. Greubel, Dec 28 2017
  • Mathematica
    r = 1 + 2^(1/2);
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190180 *)
RealDigits[N[%%, 120]]     (* A190179 *)
N[%%%, 40]
ContinuedFraction[(1 + Sqrt[-3 + 4*Sqrt[2]])/2, 100] (* G. C. Greubel, Dec 28 2017 *)
  • PARI
    contfrac((1+sqrt(-3+4*sqrt(2)))/2) \\ G. C. Greubel, Dec 28 2017

A241233 Number of right triangles, distinct up to congruence, on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 1, 5, 13, 25, 42, 62, 86, 115, 150, 191, 234, 282, 334, 395, 455, 526, 601, 677, 762, 855, 947, 1045, 1152, 1261, 1378, 1498, 1619, 1757, 1900, 2041, 2176, 2334, 2507, 2661, 2838, 3011, 3174, 3379, 3577, 3773, 3967, 4179, 4389, 4618, 4848, 5090, 5311, 5559, 5792, 6068
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 right triangles is the following:
/. *
* . *
\. .

Links

Crossrefs

Cf. A189979, A241225.

Formula

a(n) = A241231(n) - A241232(n) - A241234(n).

Extensions

a(7) from Martin Renner, May 31 2014
a(8)-a(21) from Giovanni Resta, May 31 2014
More terms from Bert Dobbelaere, Oct 17 2022
Showing 1-5 of 5 results.

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