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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A120063 Shortest side c of all integer-sided triangles with sides a<=b<=c and inradius n.

Original entry on oeis.org

5, 10, 12, 15, 25, 24, 35, 30, 36, 39, 55, 45, 65, 63, 53, 60, 85, 68, 95, 75, 77, 88, 115, 85, 125, 130, 108, 105, 145, 106, 155, 120, 132, 170, 137, 135, 185, 190, 156, 150, 205, 154, 215, 165, 159, 230, 235, 170, 245, 195, 204, 195, 265, 204, 200, 195, 228, 290
Offset: 1

Views

Author

Hugo Pfoertner, Jun 13 2006

Keywords

Comments

Terms a(11),..., a(100) computed by Thomas Mautsch (mautsch(AT)ethz.ch).
Empirically, 2*sqrt(3) < a(n)/n <= 5. The lower bound is provably tight, the upper bound seems to be achieved infinitely often, e.g, for prime n >= 5. It appears that a(p) = 5p for prime p != 3. - David W. Wilson, Jun 17 2006
Minimum of longest side occurring among all A120062(n) triangles having integer sides with integer inradius n.

Examples

			a(1)=5 because the only triangle with integer sides and inradius 1 is {3,4,5}; its longest side is 5.
a(2)=10: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their longest sides is min(13,10,29,20,17)=10.

References

  • Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

Crossrefs

See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A120062 [triangles with integer inradius], A120252 [primitive triangles with integer inradius], A057721 [maximum of longest sides], A058331 [maximum of shortest sides], A120064 [minimum of middle sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A007237.

A120570 Smallest perimeter of triangles with integer sides and inradius n.

Original entry on oeis.org

12, 24, 32, 42, 60, 64, 84, 84, 96, 108, 132, 126, 156, 162, 156, 168, 204, 190, 228, 210, 220, 240, 276, 250, 300, 312, 288, 294, 348, 312, 372, 336, 352, 408, 372, 378, 444, 456, 416, 420, 492, 440, 516, 462, 468, 552, 564, 500, 588, 540, 544, 546, 636, 570
Offset: 1

Views

Author

David W. Wilson, Jun 17 2006

Keywords

Comments

Empirically, 6*sqrt(3) <= a(n)/n <= 12. The lower bound is provably tight.
a(n) == 0 (mod 4).

References

  • Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

Crossrefs

See A120062 for sequences related to integer-sided triangles with integer inradius n.

A331043 a(n) is the number of triangles with integer sides i <= j <= k with squared radius of incircle b(n) = A331040(n)/A331041(n). Records of numbers of distinct triangles such that all smaller radii produce fewer triangles sharing the same radius of incircle than the current radius b(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 13, 14, 20, 24, 42, 45, 50, 68, 72, 84, 88, 101, 120, 149, 175, 181, 189, 206, 243, 289
Offset: 1

Views

Author

Hugo Pfoertner, Jan 11 2020

Keywords

Comments

See A331040 for more information and examples.

Crossrefs

Cf. A120062, A331040, A331041, A331042.

A120064 Shortest side b of all integer-sided triangles with sides a<=b<=c and inradius n.

Original entry on oeis.org

4, 8, 10, 14, 20, 20, 28, 28, 30, 39, 44, 40, 52, 56, 50, 56, 68, 60, 76, 70, 70, 87, 92, 80, 100, 100, 90, 97, 116, 100, 124, 112, 110, 136, 120, 120, 148, 152, 130, 140, 164, 140, 172, 154, 150, 184, 188, 160, 196, 174, 170, 182, 212, 180, 196, 189, 190, 232, 236
Offset: 1

Views

Author

Hugo Pfoertner, Jun 13 2006

Keywords

Comments

Terms a(11),..., a(100) computed by Thomas Mautsch (mautsch(AT)ethz.ch).

Examples

			a(1)=2 because the only triangle with integer sides a<=b<c and inradius 1 is {3,4,5}; its middle side is 4.
a(2)=8: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their middle sides is min(12,8,25,15,10)=8.

References

  • Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

Crossrefs

Cf. A120062 [triangles with integer inradius], A120252 [primitive triangles with integer inradius], A057721 [maximum of longest sides], A120063 [minimum of longest sides], A058331 [maximum of shortest sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A007237.
See A120062 for sequences related to integer-sided triangles with integer inradius n.

A120573 a(n) = n^5 + 3n^3 + 2n = n(n^2+1)(n^2+2).

Original entry on oeis.org

6, 60, 330, 1224, 3510, 8436, 17850, 34320, 61254, 103020, 165066, 254040, 377910, 546084, 769530, 1060896, 1434630, 1907100, 2496714, 3224040, 4111926, 5185620, 6472890, 8004144, 9812550, 11934156, 14408010, 17276280, 20584374, 24381060
Offset: 1

Views

Author

David W. Wilson, Jun 17 2006

Keywords

Comments

Largest area of any triangle with integer sides a <= b <= c and inradius n. Triangle has sides (n^2+2, n^4+2n^2+1, n^4+3n^2+1).
a(n) = A002522(n)*A054602(n). - Zerinvary Lajos, Apr 20 2008

Links

Crossrefs

See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A002522, A054602.

Programs

  • Maple
    with(combinat):seq(lcm(fibonacci(4,n),fibonacci(3,n)),n=1..30); # Zerinvary Lajos, Apr 20 2008
  • Mathematica
    LinearRecurrence[{6,-15,20,-15,6,-1},{6,60,330,1224,3510,8436},30] (* Harvey P. Dale, Aug 14 2023 *)

A331042 a(n) = 4 * squared radius of inscribed circles of triangles with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. If this radius is not a multiple of (1/4), a(n) = 0.

Original entry on oeis.org

0, 0, 3, 7, 12, 15, 32, 35, 55, 63, 95, 119, 135, 224, 231, 255, 320, 351, 455, 495, 855, 864, 896, 1071, 1440
Offset: 1

Views

Author

Hugo Pfoertner, Jan 11 2020

Keywords

Comments

It is conjectured that all radii of incircles leading to records with the exception of the first two terms are multiples of 1/4, thus a(n) > 0 for all n > 2.
See A331040 for more information and examples.

Crossrefs

Cf. A120062, A331012.
Cf. A331040, A331041, A331043 (records of numbers of triangles).

Formula

If A331041(n) equals 1 or 4, a(n) = 4 * A331040(n)/A331041(n), 0 otherwise.

A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128, 129, 132, 135
Offset: 1

Views

Author

Bernard Schott, May 05 2023

Keywords

Comments

The inradius for isosceles triangle (a, a, c) is r = (c/2)*sqrt((2*a-c)/(2*a+c)).
If m is a term, so is k*m with k > 1.
As r = 3 and r = 4 are terms, A008585 and A008586 are respective subsequences; the only terms < 100 that are not multiples of 3 or 4 are 35 and 70, the next one is r = 154 = 2*7*11 for triple (765, 765, 1386).
By the triangle inequality, a+1 <= c <= 2*a-1.
Differs from A059267. Examples: 154 is not in A059267 but in this sequence at radius r=154 with side lengths c=1386 and a=765. 442 is not in A059267 but in this sequences with r=442, c=6630, a=3435. - R. J. Mathar, Jun 26 2023

Examples

			The smallest inradius r = 3 corresponds to isosceles triangle (10, 10, 12).
The second inradius r = 4 corresponds to isosceles triangle (15, 15, 24).
r = 15 is the first inradius for which there exist two such isosceles triangles: (50, 50, 60) and (68, 68, 120).
r = 35 is the smallest inradius that is not multiple of 3 or of 4, this inradius corresponds to isosceles triangle (222, 222, 420).

Links

Crossrefs

Cf. A008585, A008586, A070204, A120062, A120570.
Cf. A362669 (similar but with (a,b,b)).

A120261 Number of primitive triangles with integer sides a<=b<=c and inradius n; primitive means gcd(a, b, c) = 1.

Original entry on oeis.org

1, 4, 10, 11, 13, 28, 17, 26, 31, 31, 20, 77, 28, 46, 67, 40, 28, 100, 26, 72, 120, 62, 32, 139, 44, 53, 71, 118, 32, 202, 35, 70, 135, 73, 97, 211, 33, 80, 130, 134, 36, 284, 45, 141, 183, 78, 50, 226, 68, 112, 150, 146, 38, 173, 150, 219, 182, 80, 38, 468, 36, 82
Offset: 1

Views

Author

David W. Wilson, Jun 13 2006

Keywords

Examples

			a(3)=10 because 10 triangles have coprime integer sides and inradius 3, namely (7,24,25) (7,65,68) (8,15,17) (11,13,20) (12,55,65) (13,40,51) (15,28,41) (16,25,39) (19,20,37) (11,100,109).

References

  • Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

Crossrefs

Cf. A120062, A120252.
See A120062 for sequences related to integer-sided triangles with integer inradius n.

A120571 2n^4+6n^2+4 = 2(n^2+1)(n^2+2).

Original entry on oeis.org

12, 60, 220, 612, 1404, 2812, 5100, 8580, 13612, 20604, 30012, 42340, 58140, 78012, 102604, 132612, 168780, 211900, 262812, 322404, 391612, 471420, 562860, 667012, 785004, 918012, 1067260, 1234020, 1419612, 1625404, 1852812, 2103300, 2378380
Offset: 1

Views

Author

David W. Wilson, Jun 17 2006

Keywords

Comments

Largest perimeter of any triangle with integer sides a<=b<=c and inradius n. Triangle has sides (n^2+2,n^4+2n^2+1,n^4+3n^2+1).

Links

Crossrefs

See A120062 for sequences related to integer-sided triangles with integer inradius n.

Programs

Formula

From Chai Wah Wu, Apr 15 2017: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
G.f.: x*(-4*x^4 + 8*x^3 - 40*x^2 - 12)/(x - 1)^5. (End)

A362669 Integer inradii for which there exists an isosceles triangle with integer sides (a, b, b) where a < b.

Original entry on oeis.org

10, 20, 21, 24, 30, 36, 40, 42, 48, 50, 55, 60, 63, 70, 72, 78, 80, 84, 90, 96, 100, 105, 108, 110, 112, 120, 126, 130, 136, 140, 144, 147, 150, 156, 160, 165, 168, 170, 171, 180, 189, 190, 192, 195, 200, 210, 216, 220, 224, 230, 231, 234, 240, 250, 252, 253, 260, 264, 270, 272, 273, 275
Offset: 1

Views

Author

Bernard Schott, Apr 29 2023

Keywords

Comments

The inradius for isosceles triangle (a, b, b) is r = (a/2)*sqrt((2*b-a)/(2*b+a)).
If m is a term, so is k*m with k > 1; hence, A008592 \ {0} is a subsequence.

Examples

			The smallest inradius, r = 10, corresponds to isosceles triangle (30, 39, 39).
The third inradius, r = 21, corresponds to isosceles triangle (56, 100, 100).
r = 60 is the first inradius for which there exist two such isosceles triangles: (168, 259, 259) and (180, 234, 234).

Links

Crossrefs

Cf. A008592, A070204, A120062, A120570, A362670 (similar but with (a,a,c)).

Programs

  • Mathematica
    Select[Range[300], Length @ Reduce[#^2 == a^2*(2*b - a)/(4*(2*b + a)) && 0 < a < b, {a, b}, Integers] > 0 &] (* Amiram Eldar, May 05 2023 *)
Previous Showing 11-20 of 26 results. Next

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