A331251 -id:A331251 - cp's OEIS frontend

A331252 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives middle side j. The other sides are in A331251 and A331253.

1, 2, 3, 2, 4, 2, 5, 3, 3, 6, 7, 4, 4, 3, 8, 3, 3, 9, 5, 5, 10, 4, 11, 6, 4, 6, 12, 4, 7, 13, 5, 4, 7, 4, 14, 5, 8, 5, 15, 6, 4, 8, 4, 16, 9, 5, 17, 6, 7, 9, 6, 18, 10, 5, 19, 6, 5, 5, 11, 8, 5, 10, 20, 7, 7, 21, 12, 5, 7, 11, 22, 9, 6, 6, 23, 6, 13, 8, 8, 12

Offset: 1

Hugo Pfoertner, Jan 19 2020

nonn

			See A331251.

Cf. A331251 (shortest side), A331253 (longest side).

A331253 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives longest side k. The other sides are in A331251 and A331252.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 3, 4, 6, 7, 5, 4, 3, 8, 5, 4, 9, 6, 5, 10, 6, 11, 7, 4, 6, 12, 5, 8, 13, 7, 7, 7, 4, 14, 5, 9, 6, 15, 8, 5, 8, 6, 16, 10, 8, 17, 6, 9, 9, 7, 18, 11, 5, 19, 9, 7, 9, 12, 10, 6, 10, 20, 7, 8, 21, 13, 5, 10, 11, 22, 11, 6, 10, 23, 8, 14, 8, 9

Offset: 1

Hugo Pfoertner, Jan 19 2020

nonn

			See A331251.

Cf. A331251 (shortest side), A331252 (middle side).

A070080 Smallest side of integer triangles [a(n) <= A070081(n) <= A070082(n)], sorted by perimeter, lexicographically ordered.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 3, 2, 3, 1, 2, 3, 3, 2, 3, 4, 1, 2, 3, 3, 4, 2, 3, 4, 4, 1, 2, 3, 3, 4, 4, 5, 2, 3, 4, 4, 5, 1, 2, 3, 3, 4, 4, 5, 5, 2, 3, 4, 4, 5, 5, 6, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 2, 3, 4, 4, 5, 5, 6, 6, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 2, 3, 4, 4, 5, 5

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

G. C. Greubel, Table of n, a(n) for the first 55 rows
Reinhard Zumkeller, Integer-sided triangles

Cf. A046128, A055594, A069597.
Cf. A070081, A070082, A070083, A070084, A070085, A070086.
Cf. A316841, A316843, A316844, A316845 (sides (i,j,k) with j + k > i >= j >= k >= 1).
Cf. A331244, A331245, A331246 (similar, but triangles sorted by radius of enclosing circle), A331251, A331252, A331253 (triangles sorted by area), A331254, A331255, A331256 (triangles sorted by radius of circumcircle).

m = 55 (* 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]]&];
triangles[[All, 1]] (* Jean-François Alcover, Jun 12 2012, updated Jul 09 2017 *)

a(n) = A070083(n) - A070082(n) - A070081(n).

A010466 Decimal expansion of square root of 8.

Original entry on oeis.org

2, 8, 2, 8, 4, 2, 7, 1, 2, 4, 7, 4, 6, 1, 9, 0, 0, 9, 7, 6, 0, 3, 3, 7, 7, 4, 4, 8, 4, 1, 9, 3, 9, 6, 1, 5, 7, 1, 3, 9, 3, 4, 3, 7, 5, 0, 7, 5, 3, 8, 9, 6, 1, 4, 6, 3, 5, 3, 3, 5, 9, 4, 7, 5, 9, 8, 1, 4, 6, 4, 9, 5, 6, 9, 2, 4, 2, 1, 4, 0, 7, 7, 7, 0, 0, 7, 7, 5, 0, 6, 8, 6, 5, 5, 2, 8, 3, 1, 4, 5, 4, 7

Offset: 1

N. J. A. Sloane

Sqrt(8) = 2*sqrt(2) is the length of the longest (rigid) ladder that can be carried horizontally around a right angled corner in a hallway of unit width. - Lekraj Beedassy, Apr 19 2006
Continued fraction expansion is 2 followed by {1, 4} repeated. - Harry J. Smith, Jun 05 2009
This is the second Lagrange number. - Alonso del Arte, Dec 06 2011
Also 2*sqrt(2) is the ratio of the perimeter of a square to its diameter (diagonal length). - Rick L. Shepherd, Dec 29 2016
Uchiyama shows that every interval (n, n + c*n^(1/4)) contains an integer that is the sum of two squares, where c = 2^(3/2). - Michel Marcus, Jan 03 2018
This is the area of the eighth-smallest triangle with integer side lengths (2, 3, 3), or the seventh-smallest triangle if two smaller triangles with the same area are counted only once (see A331251). - Hugo Pfoertner, Feb 12 2020
Diameter of a sphere whose surface area equals 8*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Feb 13 2020
Sqrt(8) = area between the curves y = sin(x) and y = cos(x) for Pi/4 < x < 5 Pi/4; this is one of infinitely many congruent convex regions bounded by the two curves. - Clark Kimberling, May 03 2020
Area of the regular 8-gon with circumradius =1. - R. J. Mathar, Aug 24 2023

			2.828427124746190097603377448419396157139343750753896146353359475981464...
Sqrt(8) = sqrt(1+2*i*sqrt(2)) + sqrt(1-2*i*sqrt(2)) = sqrt(1/2+2*i*sqrt(3)) + sqrt(1/2-2*i*sqrt(3)), where i=sqrt(-1). - _Bruno Berselli_, Nov 20 2012
1 + 3/4 + 3*5/(4*8) + 3*5*7/(4*8*12) + 3*5*7*9/(4*8*12*16) + ... - _Bruno Berselli_, Mar 16 2014

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 187.
S. R. Finch, Moving Sofa Constant, Sect. 8.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 519-523, 2003.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Jason Kimberley, Index of expansions of sqrt(d) in base b
R. J. Nemiroff & J. Bonnell, The first 1 million digits of the square root of 8
R. J. Nemiroff & J. Bonnell, Plouffe's Inverter, The first 1 million digits of the square root of 8
Ana Rechtman, Juin 2023, 3e défi, Images des Mathématiques, CNRS, 2023.
S. Uchiyama, On the distribution of integers representable as a sum of two h-th powers, J. Fac. Sci. Hokkaido Univ. Ser. I, 18, 124-127, 1964/1965.
Eric Weisstein's World of Mathematics, Moving Ladder Problem
Index entries for algebraic numbers, degree 2

Cf. A040005 (continued fraction).

Cf. A331250, A331251.

SetDefaultRealField(RealField(100)); Sqrt(8); // Vincenzo Librandi, Feb 13 2020

evalf(2^(3/2)) ; # R. J. Mathar, Jul 15 2013

RealDigits[N[Sqrt[8],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

default(realprecision, 20080); x=sqrt(8); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010466.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

Equals 1 + Sum_{n>=1} ( Product_{k=1..n} (2k+1)/(4k) ). - Bruno Berselli, Mar 16 2014
Equals 2*A002193. - R. J. Mathar, Jan 14 2021
From Peter Bala, Mar 01 2022: (Start)
Equals 3*Sum_{n >= 0} (1/(4*n+1) - 1/(4*n-3))*binomial(1/2,n). Cf. A002580 and A175576.
Equals 4*hypergeom([-1/2, -3/4], [5/4], -1). (End)
Equals 8 * A020765. - R. J. Mathar, Aug 24 2023

A140239 Decimal expansion of 3*sqrt(15)/4.

Original entry on oeis.org

2, 9, 0, 4, 7, 3, 7, 5, 0, 9, 6, 5, 5, 5, 6, 2, 6, 6, 3, 8, 8, 4, 4, 4, 9, 0, 4, 9, 8, 3, 6, 7, 9, 9, 7, 0, 8, 1, 2, 4, 6, 9, 1, 2, 7, 8, 9, 6, 8, 6, 9, 3, 1, 1, 9, 9, 4, 0, 6, 8, 0, 3, 2, 4, 5, 8, 5, 1, 1, 2, 3, 1, 8, 9, 5, 2, 7, 3, 4, 2, 7, 5, 1, 3, 9, 4, 6, 5, 5, 3, 2, 6, 4, 4, 0, 0, 5, 1, 3, 8, 4, 3, 7, 2, 2

Offset: 1

Rick L. Shepherd, May 14 2008

Area of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths.

This is the area of the ninth-smallest triangle with integer side lengths, or the eighth-smallest triangle if two smaller triangles with the same area are counted only once (see A331251). - Hugo Pfoertner, Feb 12 2020

			2.90473750965556266388444904983679970812469127896869311994068032458511231895...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. A140240, A140241, A140242, A140243, A140244, A140245, A140246, A140247, A140248, A140249, A088543, A010472, A331250, A331251.

RealDigits[(3Sqrt[15])/4,10,120][[1]] (* Harvey P. Dale, Apr 03 2013 *)

3*sqrt(15)/4 = 3*A010472/4.

A331244 Triangles with integer sides i <= j <= k sorted by radius of enclosing circle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331245 and A331246.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 3, 1, 2, 3, 4, 2, 3, 3, 1, 2, 4, 3, 4, 5, 2, 3, 3, 4, 1, 4, 2, 3, 5, 4, 5, 6, 2, 3, 3, 4, 4, 5, 4, 1, 2, 5, 3, 4, 6, 5, 6, 2, 3, 3, 4, 4, 5, 5, 4, 1, 6, 2, 5, 7, 3, 4, 6, 5, 7, 6, 7, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 1, 5, 2, 3, 7, 6, 4

Offset: 1

Hugo Pfoertner, Jan 20 2020

nonn

The enclosing circle differs from the circumcircle by limiting the radius to (longest side)/2 for obtuse triangles, i.e., those with i^2 + j^2 < k^2.

			List of triangles begins:
   n
   |     R^2
   |     |    i .... (this sequence)
   |     |    | j .. (A331245)
   |     |    | | k  (A331246)
   |     |    | | |
   1    1/ 3  1 1 1
   2   16/15  1 2 2
   3    4/ 3  2 2 2
   4    9/ 4  2 2 3  obtuse
   5   81/35  1 3 3
   6   81/32  2 3 3
   7    3/ 1  3 3 3
   8    4/ 1  2 3 4  obtuse
   9   81/20  3 3 4
  10  256/63  1 4 4
  11   64/15  2 4 4
  12  256/55  3 4 4
  13   16/ 3  4 4 4
  14   25/ 4  2 4 5  obtuse
  15   25/ 4  3 3 5  obtuse
  16   25/ 4  3 4 5
  17  625/99  1 5 5

Cf. A128006, A128007, A192493, A192494, A331227, A331228, A331251, A331252, A331253, A331254, A331255, A331256.

Cf. A331245 (middle side), A331246 (longest side).

A331250 a(n) = number of triangles with integer sides i <= j <= k with area <= n.

Original entry on oeis.org

2, 6, 10, 15, 21, 28, 35, 44, 52, 63, 71, 84, 92, 105, 118, 128, 143, 159, 173, 183, 200, 214, 231, 248, 264, 280, 301, 316, 332, 356, 370, 394, 414, 428, 451, 475, 494, 514, 535, 557, 580, 607, 624, 645, 678, 697, 718, 748, 770, 794, 822, 845, 873, 900, 927

Offset: 1

Hugo Pfoertner, Jan 20 2020

nonn

			The sorted list of areas A_k = A(A331251(k), A331252(k), A331253(k)), rounded to 10^-4, starts:: {0.43301, 0.96825, 1.4790, 1.7321, 1.9843, 1.9843, 2.4875, 2.8284, 2.9047, 2.9896, 3.4911, 3.7997, 3.8730, 3.8971, 3.9922, 4.1458, 4.4721, 4.4931, 4.6837, 4.8990, 4.9937, 5.3327, ...}.
a(1) = 2: 2 triangles (A = 0.43301, 0.96825) with A <= 1,
a(2) = 6: a(1) + 4 triangles (A = 1.4790, 1.7321, 1.9843, 1.9843) with 1 < A <= 2,
a(3) = 10: a(2) + 4 triangles (A = 2.4875, 2.8284, 2.9047, 2.9896) with 2 < A <= 3,
a(4) = 15: a(3) + 5 triangles (A = 3.4911, 3.7997, 3.8730, 3.8971, 3.9922) with 3 < A <= 4,
a(5) = 21: a(4) + 6 triangles (A = 4.1458, 4.4721, 4.4931, 4.6837, 4.8990, 4.9937) with 4 < A <= 5.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A316853, A317182, A331011, A331251, A331252, A331253.

from itertools import count
def A331250(n):
    m, c = n**2<<4, 0
    for k in count(1):
        if (k**2<<2) - 1 > m:
            break
        for j in range((k>>1)+1,k+1):
            for i in range(k-j+1,j+1):
                if ((-i + j + k)*(i - j + k)*(i + j - k)*(i + j + k)) > m:
                    break
                c += 1
    return c # Chai Wah Wu, Aug 25 2023

Area A of a triangle with sides a, b, c:

A(a, b, c) = sqrt(s*(s - a)*(s - b)*(s - c)) with s = (a + b + c)/2.

A331254 Triangles with integer sides i <= j <= k sorted by radius of circumcircle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331255 and A331256.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 3, 1, 2, 2, 3, 4, 3, 1, 2, 4, 3, 2, 3, 4, 5, 3, 1, 4, 4, 2, 3, 5, 4, 2, 5, 3, 6, 5, 4, 1, 3, 2, 4, 5, 3, 4, 6, 5, 2, 6, 4, 5, 1, 6, 2, 3, 3, 5, 7, 3, 4, 4, 4, 6, 5, 5, 7, 6, 2, 7, 4, 6, 1, 5, 5, 2, 6, 3, 3, 7, 6, 4, 8, 5, 4, 7, 3, 5, 6, 8

Offset: 1

Hugo Pfoertner, Jan 19 2020

nonn

			List of triangles begins:
   n
   |     R^2 = A331227(n)/A331228(n)
   |     |    i .... (this sequence)
   |     |    | j .. (A331255)
   |     |    | | k  (A331256)
   |     |    | | |
   1    1/ 3  1 1 1
   2   16/15  1 2 2
   3    4/ 3  2 2 2
   4   16/ 7  2 2 3
   5   81/35  1 3 3
   6   81/32  2 3 3
   7    3/ 1  3 3 3
   8   81/20  3 3 4
   9  256/63  1 4 4
  10   64/15  2 3 4
  11   64/15  2 4 4
  12  256/55  3 4 4

Cf. A331227, A331228, A331251, A331252, A331253.

Cf. A331255 (middle side), A331256 (longest side).

A331247 Numerator of the x-coordinate of the 3rd point (x,y) of the n-th triangle with integer sides i <= j <= k in a list sorted by increasing area, when the triangle is drawn with the shortest side from (0,0) to (0,i) and the middle side from (0,i) to (x,y). x = a(n)/A331248(n), y = sqrt(A331249(n))/A331248(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 1, 1, 11, 1, 1, 13, 1, 3, 1, 25, 8, 1, 15, 1, 1, 29, 1, 17, 3, 1, 1, 3, 19, 1, 11, 49, 1, 2, 1, 3, 21, 10, 1, 37, 25, 1, 9, 1, 23, 55, 1, 3, 41, 1, 11, 1, 25, 2, 1, 61, 5, 81, 27, 15, 27, 1, 1, 3, 4, 1, 29, 5, 67, 1, 1, 49, 2, 89, 1, 11, 31, 3

Offset: 1

Hugo Pfoertner, Jan 23 2020

The side lengths (i,j,k) of the triangles in the list sorted by area are given in A331251, A331252, and A331253.

			x(n) = a(n) / A331248(n),
y(n) = sqrt(A331249(n)) / A331248(n),
   n i (A331251)
   | | j (A331252)
   | | | k (A331253)
   | | | |    A^2*16
   | | | |    |  a(n) this sequence
   | | | |    |  |  A331248
   | | | |    |  |  |   A331249
   | | | |    |  |  |   |  (x, y)
   1 1 1 1    3  1  2   3  (0.5000, 0.86603)
   2 1 2 2   15  1  2  15  (0.5000, 1.9365)
   3 1 3 3   35  1  2  35  (0.5000, 2.9580)
   4 2 2 2   48  1  1   3  (1.0000, 1.7321)
   5 1 4 4   63  1  2  63  (0.5000, 3.9686)
   6 2 2 3   63  9  4  63  (2.2500, 1.9843)
   7 1 5 5   99  1  2  99  (0.5000, 4.9749)
   8 2 3 3  128  1  1   8  (1.0000, 2.8284)
   9 2 3 4  135 11  4 135  (2.7500, 2.9047)
  10 1 6 6  143  1  2 143  (0.5000, 5.9791)
  11 1 7 7  195  1  2 195  (0.5000, 6.9821)
  12 2 4 5  231 13  4 231  (3.2500, 3.7997)
  13 2 4 4  240  1  1  15  (1.0000, 3.8730)
  14 3 3 3  243  3  2  27  (1.5000, 2.5981)
  ...
  28 3 4 5  576  3  1  16  (3.0000, 4.0000)

Cf. A331248, A331249, A331251, A331252, A331253, A331695, A331696, A331697.

A331248 Common denominator of x-coordinate and y-coordinate of 3rd point of triangles with integer sides corresponding to A331247 and A331249.

Original entry on oeis.org

2, 2, 2, 1, 2, 4, 2, 1, 4, 2, 2, 4, 1, 2, 2, 6, 3, 2, 4, 1, 2, 6, 2, 4, 2, 1, 2, 1, 4, 2, 2, 8, 1, 1, 2, 2, 4, 3, 2, 6, 8, 1, 2, 2, 4, 8, 2, 2, 6, 1, 3, 2, 4, 1, 2, 8, 1, 10, 4, 2, 8, 1, 2, 2, 1, 2, 4, 2, 8, 1, 2, 6, 1, 10, 2, 2, 4, 2, 3, 1, 8, 2, 5, 5, 6, 8, 2

Offset: 1

Hugo Pfoertner, Jan 23 2020

			See A331247.

Cf. A331247, A331249, A331251, A331252, A331253.

cp's OEIS Frontend

A331252 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives middle side j. The other sides are in A331251 and A331253.

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A331253 Triangles with integer sides i <= j <= k sorted by area, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives longest side k. The other sides are in A331251 and A331252.

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A070080 Smallest side of integer triangles [a(n) <= A070081(n) <= A070082(n)], sorted by perimeter, lexicographically ordered.

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A010466 Decimal expansion of square root of 8.

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A140239 Decimal expansion of 3*sqrt(15)/4.

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A331244 Triangles with integer sides i <= j <= k sorted by radius of enclosing circle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331245 and A331246.

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A331250 a(n) = number of triangles with integer sides i <= j <= k with area <= n.

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Python

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A331254 Triangles with integer sides i <= j <= k sorted by radius of circumcircle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331255 and A331256.

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A331248 Common denominator of x-coordinate and y-coordinate of 3rd point of triangles with integer sides corresponding to A331247 and A331249.

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