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 1-10 of 10 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 *)

A070101 Number of obtuse integer triangles with perimeter n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 3, 2, 3, 3, 5, 3, 7, 4, 8, 5, 9, 7, 10, 8, 11, 9, 14, 11, 16, 12, 18, 14, 19, 17, 21, 18, 23, 21, 27, 22, 30, 24, 32, 27, 34, 30, 37, 33, 40, 35, 44, 37, 47, 40, 50, 44, 53, 49, 56, 52, 60, 55, 64, 57, 68
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

An integer triangle [A070080(k) <= A070081(k) <= A070082(k)] is obtuse iff A070085(k) < 0.

Examples

			For n=14 there are A005044(14)=4 integer triangles: [2,6,6], [3,5,6], [4,4,6] and [4,5,5]; two of them are obtuse, as 3^2+5^2<36=6^2 and 4^2+4^2<36=6^2, therefore a(14)=2.

Links

Crossrefs

Cf. A070102, A070103, A070127.
Cf. A005044, A024155, A070093.

Programs

  • Mathematica
    Table[Sum[Sum[(1 - Sign[Floor[(i^2 + k^2)/(n - i - k)^2]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 12 2019 *)

Formula

a(n) = A005044(n) - A070093(n) - A024155(n).
a(n) = A024156(n) + A070106(n).
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)}
(1-sign(floor((i^2 + k^2)/(n-i-k)^2))) * sign(floor((i+k)/(n-i-k+1))). - Wesley Ivan Hurt, May 12 2019

A070131 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

8, 13, 20, 21, 25, 29, 30, 36, 37, 41, 42, 44, 49, 56, 57, 59, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 86, 89, 96, 97, 99, 100, 101, 102, 105, 110, 111, 113, 115, 122, 123, 125, 126, 127, 128, 130, 131, 138, 141, 144, 147, 152, 153
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(20)=69: [A070080(69), A070081(69), A070082(69)]=[5<6<9], A070084(69)=gcd(5,6,9)=1, A070085(69)=5^2+6^2-9^2=25+36-81=-20<0.

Links

Crossrefs

Cf. A070104, A070110, A070127, A070112.

A070132 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle with prime side lengths.

Original entry on oeis.org

30, 101, 153, 193, 328, 334, 392, 444, 454, 519, 603, 621, 771, 777, 795, 878, 1005, 1123, 1135, 1508, 1526, 1538, 1694, 1818, 1848, 1858, 1999, 2023, 2037, 2066, 2193, 2223, 2253, 2454, 2663, 2903, 3055, 3321, 3363
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=519: [A070080(519), A070081(519), A070082(519)]=[5<17<19], A070085(519)=5^2+17^2-19^2=25+289-361=-47<0.

Links

Crossrefs

Cf. A070105, A070111, A070127, A070112.

A070134 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

5, 14, 32, 52, 61, 104, 118, 133, 146, 163, 202, 242, 246, 266, 314, 342, 404, 437, 467, 472, 504, 542, 547, 577, 619, 625, 714, 757, 801, 807, 853, 907, 957, 1015, 1022, 1082, 1139, 1145, 1265, 1278, 1335, 1414, 1475
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=61: [A070080(61), A070081(61), A070082(61)]=[5=5<9], A070084(69)=gcd(5,5,9)=1, A070085(61)=5^2+5^2-9^2=25+25-81=-31<0.

Links

Crossrefs

Cf. A070107, A070110, A070127, A070115.

A070135 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with prime side lengths.

Original entry on oeis.org

5, 14, 133, 163, 472, 542, 714, 907, 1888, 2064, 2267, 2466, 3133, 4240, 4858, 5188, 7563, 8469, 9466, 9958, 11069, 12841, 13471, 14876, 17032, 17794, 20214, 20268, 21125, 21943, 22843, 22933, 23837, 25767, 25797
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(3)=133: [A070080(133), A070081(133), A070082(133)]=[7=7<11], A070085(133)=7^2+7^2-11^2=49+49-121=-23<0.

Links

Crossrefs

Cf. A070108, A070111, A070127, A070115.

A070128 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse integer triangle with relatively prime side lengths.

Original entry on oeis.org

5, 8, 13, 14, 20, 21, 25, 29, 30, 32, 36, 37, 41, 42, 44, 49, 52, 56, 57, 59, 61, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 86, 89, 96, 97, 99, 100, 101, 102, 104, 105, 110, 111, 113, 115, 118, 122, 123, 125, 126, 127, 128, 130, 131, 133
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(9)=30: [A070080(30), A070081(30), A070082(30)]=[3,5,7], A070084(30)=gcd(3,5,7)=1, A070085(30)=3^2+5^2-7^2=9+25-49=-15>0.

Links

Crossrefs

Cf. A070102, A070131, A070134, A070110, A070127.

A070129 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse integer triangle with prime side lengths.

Original entry on oeis.org

5, 14, 30, 101, 133, 153, 163, 193, 328, 334, 392, 444, 454, 472, 519, 542, 603, 621, 714, 771, 777, 795, 878, 907, 1005, 1123, 1135, 1508, 1526, 1538, 1694, 1818, 1848, 1858, 1888, 1999, 2023, 2037, 2064, 2066, 2193
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(3)=30: [A070080(30), A070081(30), A070082(30)]=[3,5,7], A070085(30)=3^2+5^2-7^2=9+25-49=-15>0.

Links

Crossrefs

Cf. A070103, A070132, A070135, A070111, A070127.

A070133 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle.

Original entry on oeis.org

5, 14, 26, 32, 52, 61, 82, 91, 104, 118, 133, 146, 163, 182, 202, 219, 242, 246, 266, 291, 314, 342, 347, 372, 404, 432, 437, 467, 472, 504, 542, 547, 577, 582, 619, 625, 663, 709, 714, 751, 757, 801, 807, 853, 858, 907, 913
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=52: [A070080(52), A070081(52), A070082(52)]=[5=5<8], A070085(52)=5^2+5^2-8^2=25+25-64=-14<0.

Links

Crossrefs

Cf. A070106, A070134, A070135, A070115, A070127.

A144609 Sturmian word of slope Pi.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jan 13 2009

Keywords

Comments

A063438 seems to contain the run lengths of 1's. - R. J. Mathar, May 30 2025

Crossrefs

See A144595 for further details.
Seems to be very similar to A070127. Is this a coincidence?
Cf. A063438, A076539 (partial sums).

Programs

  • Maple
    Digits := 500 :
x :=1 ;
y :=0 ;
slop := Pi ;
printf("0,") ;
for n from 1 to 300 do
    if evalf((y+1)/x-slop) > 0 then
        x := x+1 ;
        printf("0,") ;
    else
        y := y+1 ;
        printf("1,") ;
    end if;
end do: # R. J. Mathar, May 30 2025
  • Mathematica
    christoffel[s_, M_] := Module[{n, x = 1, y = 0, ans = {0}}, Do[ If[y + 1 <= s*x, AppendTo[ans, 1]; y++, AppendTo[ans, 0]; x++], {n, 1, M}]; ans]; christoffel[Pi, 105] (* Robert G. Wilson v, Feb 02 2017, after Jean-François Alcover, Sep 19 2016, A274170 *)
Showing 1-10 of 10 results.

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

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