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.

A070085 a(n) = A070080(n)^2 + A070081(n)^2 - A070082(n)^2.

Original entry on oeis.org

1, 1, 4, 1, -1, 4, 1, -3, 9, 4, 2, 1, -5, -7, 9, 4, 0, 16, 1, -7, -11, 9, 7, 4, -2, -4, 16, 1, -9, -15, 9, -17, 5, 25, 4, -4, -8, 16, 14, 1, -11, -19, 9, -23, 3, 1, 25, 4, -6, -12, 16, -14, 12, 36, 1, -13, -23, 9, -29, 1, -31, -3, 25, 23, 4, -8
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

The integer triangle [A070080(n)<=A070081(n)<=A070082(n)] is acute iff a(n)>0, right iff a(n)=0 and obtuse iff a(0)<0.

Links

Crossrefs

Cf. A070093, A070101, A024155, A070118, A070127, A070136.

Programs

  • Mathematica
    maxPer = m = 22;
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[Ceiling[ per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2&];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
#[[1]]^2 + #[[2]]^2 - #[[3]]^2& /@ triangles (* Jean-François Alcover, Jul 31 2018 *)

A070142 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer area.

Original entry on oeis.org

17, 39, 52, 116, 212, 252, 269, 368, 370, 372, 375, 493, 561, 587, 659, 839, 850, 862, 957, 972, 1156, 1186, 1196, 1204, 1297, 1582, 1599, 1629, 1912, 1920, 1955, 1971, 1988, 2115, 2352, 2555, 2574, 2713, 2774, 2778, 2790
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(2)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Links

Crossrefs

Cf. A070088, A070149, A070143, A070144, A070145, A051516, A069594, A069597, A070136, A070137, A070209.

Programs

  • Mathematica
    maxPerim = 100; maxSide = Floor[(maxPerim - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPerim^3 + a*maxPerim^2 + b*maxPerim + c; triangles = Reap[ Do[ If[ a + b + c <= maxPerim && c - b < a < c + b && b - a < c < b + a && c - a < b < c + a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; stri = Sort[ triangles, order[#1] < order[#2]&]; area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p*(p - a)*(p - b)*(p - c)]]; Position[ stri, tri_ /; IntegerQ[area[tri]]] // Flatten (* Jean-François Alcover, Feb 22 2013 *)

A070137 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a right integer triangle with relatively prime side lengths.

Original entry on oeis.org

17, 212, 493, 1297, 2574, 4298, 5251, 14414, 16365, 21231, 26125, 39056, 42597, 55042, 63770, 75052, 91121, 97256, 124355, 164640, 200999, 213083, 253721, 275999, 367997, 384154, 415778, 478343, 511633, 518370, 606417, 665040, 689356, 755435, 846571
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

Right integer triangles have integer areas: see A070143.

Examples

			493 is a term: [A070080(493), A070081(493), A070082(493)]=[8,15,17], A070084(493)=gcd(8,15,17)=1, A070085(493)=8^2+15^2-17^2=64+225-289=0.

Links

Crossrefs

Cf. A070109, A070136.

Programs

  • Mathematica
    m = 500 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, b_, c_} /; GCD[a, b, c] == 1 && a^2 + b^2 - c^2 == 0] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

Extensions

More terms from Jean-François Alcover, Oct 04 2021

A070146 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle with integer area.

Original entry on oeis.org

39, 269, 375, 587, 862, 972, 1196, 1955, 1988, 2352, 2555, 2796, 3818, 4319, 4406, 5378, 6522, 6808, 6880, 6890, 6921, 7234, 7360, 8193, 9159, 9207, 10272, 14545, 15004, 15061, 15101, 15216, 15237, 15943, 16502
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(1)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6]: A070085(39)=5^2+5^2-6^2=14>0 and area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Links

Crossrefs

Cf. A070142, A070136, A070147, A070140.

A070147 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an obtuse integer triangle with integer area.

Original entry on oeis.org

52, 252, 368, 372, 561, 659, 839, 957, 1156, 1186, 1204, 1582, 1912, 1920, 1971, 2115, 2713, 2774, 2790, 3251, 3473, 3728, 3746, 4286, 4307, 4313, 4330, 5008, 5272, 5374, 6369, 6389, 6432, 6776, 6881, 7223, 7310, 7341
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(1)=52: [A070080(52), A070081(52), A070082(52)] = [5,5,8]: A070085(52)=5^2+5^2-8^2=-14<0 and area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(52)/2=(5+5+8)/2=9, area^2=9*4*4*1=16*9 is an integer square, therefore A070086(52)=area=4*3=12.

Links

Crossrefs

Cf. A070142, A070136, A070146, A070141.
Showing 1-5 of 5 results.

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