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.

A305717 Number of distinct Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 1, 2, 1, 3, 2, 4, 1, 3, 5, 1, 1, 2, 5, 1, 1, 4, 8, 1, 5, 5, 6, 5, 2, 12, 1, 6, 1, 5, 4, 9, 1, 9, 2, 5, 11, 8, 2, 3, 5, 7, 12, 1, 10, 1, 1, 1, 7, 10, 3, 2, 2, 15, 14, 5, 10, 5, 5, 11, 16, 5, 1, 12, 7, 3, 1, 7, 1, 2, 15, 16, 1, 11, 16, 18
Offset: 1

Views

Author

Peter Kagey, Jun 08 2018

Keywords

Examples

			For n = 7, the a(7) = 4 Heronian triangles with perimeter A051518(7) = 36 are:
the 10-13-13 isosceles triangle (area 60),
the 10-10-16 isosceles triangle (area 48),
the  9-12-15 scalene   triangle (area 54), and
the  9-10-17 scalene   triangle (area 36).

Links

Crossrefs

Cf. A051516, A051518, A070138, A298614.

Programs

  • Mathematica
    htc[p_] := Block[{t=0, c, q=p/2}, Do[c=p-a-b; If[c >= b && a+c > b && a+b > c && IntegerQ[Sqrt[q (q-a) (q-b) (q-c)]], t++], {a, p/3}, {b, a, p-a-1}]; t]; Select[htc /@ (Range[128] 2), # > 0 &] (* Giovanni Resta, Jun 14 2018 *)

Formula

a(n) = A051516(A051518(n)).

A330912 Sum of the smallest side lengths of all Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

3, 5, 5, 6, 5, 14, 38, 8, 20, 11, 37, 29, 43, 7, 31, 64, 11, 17, 37, 84, 19, 15, 70, 130, 22, 87, 101, 133, 122, 38, 241, 25, 149, 25, 111, 123, 225, 39, 220, 54, 120, 327, 254, 57, 103, 162, 227, 371, 41, 321, 34, 43, 29, 278, 373, 76, 70, 95, 577, 567, 157, 476, 221
Offset: 1

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			a(1) = 3; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its smallest side length is 3.
a(6) = 14; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum is 4 + 10 = 14.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330915, A330916.

Formula

a(n) = Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * chi(sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)))) * k, where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A330915 Sum of the "middle" side lengths (b such that a <= b <= c) of all Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

4, 5, 5, 8, 12, 23, 45, 15, 29, 13, 48, 30, 77, 24, 69, 117, 25, 25, 46, 119, 20, 26, 110, 246, 26, 167, 172, 205, 169, 79, 468, 33, 229, 38, 222, 167, 429, 41, 429, 101, 270, 560, 416, 100, 153, 276, 390, 717, 50, 615, 61, 61, 60, 404, 634, 214, 130, 130, 1033, 975, 382
Offset: 1

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			a(1) = 4; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its "middle" side length is 4.
a(6) = 23; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum is 13 + 10 = 23.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330916.

Formula

a(n) = Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * chi(sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)))) * i, where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A330916 Sum of the largest side lengths of all Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

5, 6, 8, 10, 13, 27, 61, 17, 35, 20, 59, 41, 96, 25, 80, 139, 30, 26, 57, 157, 37, 37, 140, 296, 40, 196, 207, 250, 209, 91, 587, 52, 294, 51, 267, 214, 498, 50, 539, 117, 310, 697, 530, 147, 206, 342, 503, 856, 73, 744, 75, 68, 85, 550, 793, 256, 172, 155, 1270, 1202
Offset: 1

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			a(1) = 5; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its largest side length is 5.
a(6) = 27; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum is 15 + 12 = 27.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915.

Formula

a(n) = Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * chi(sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)))) * (c(n)-i-k), where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A330921 Sum of the areas of all Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

6, 12, 12, 24, 30, 72, 198, 60, 126, 66, 288, 180, 360, 84, 330, 648, 132, 204, 420, 876, 114, 156, 840, 1764, 264, 1350, 1632, 2016, 1830, 624, 3816, 330, 2604, 456, 2280, 2352, 4800, 780, 4422, 1224, 2940, 7068, 5430, 912, 2310, 3744, 5520, 9144, 984, 8736, 1020
Offset: 1

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			a(1) = 6; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its area is 3*4/2 = 6.
a(6) = 72; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum of their areas 24 + 48 = 72.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916.

Formula

a(n) = Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * chi(sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)))) * sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k))), where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A330917 Largest possible side length, a, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

3, 5, 5, 6, 5, 10, 10, 8, 13, 11, 15, 16, 15, 7, 15, 20, 11, 17, 20, 20, 19, 15, 25, 26, 22, 25, 30, 29, 32, 25, 30, 25, 35, 25, 30, 39, 40, 39, 33, 34, 40, 45, 48, 38, 35, 51, 50, 53, 41, 52, 34, 43, 29, 55, 50, 35, 39, 57, 60, 65, 55, 64, 51, 65, 65, 60, 68, 61, 70, 65
Offset: 1

Views

Author

Wesley Ivan Hurt, May 03 2020

Keywords

Examples

			a(1) = 3; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its shortest side is 3.
a(6) = 10; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose smallest side lengths are 4 and 10. The largest of these is 10.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330923, A331199.

A330923 Largest possible side length, b, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

4, 5, 5, 8, 12, 13, 13, 15, 15, 13, 17, 17, 25, 24, 25, 29, 25, 25, 25, 29, 20, 26, 30, 35, 26, 40, 39, 40, 41, 40, 51, 33, 48, 38, 50, 45, 58, 41, 60, 51, 65, 65, 61, 60, 56, 68, 65, 75, 50, 72, 61, 61, 60, 74, 80, 84, 68, 65, 87, 89, 90, 82, 87, 80, 89, 102, 100, 74
Offset: 1

Views

Author

Wesley Ivan Hurt, May 03 2020

Keywords

Examples

			a(1) = 4; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its middle side is 4.
a(6) = 13; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose middle side lengths are 13 and 10. The largest of these is 13.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330917, A331199.

A331199 Largest possible side length, c, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

5, 6, 8, 10, 13, 15, 17, 17, 20, 20, 21, 24, 26, 25, 29, 30, 30, 26, 29, 35, 37, 37, 39, 41, 40, 41, 45, 48, 48, 51, 53, 52, 53, 51, 58, 60, 61, 50, 65, 65, 68, 70, 74, 74, 75, 75, 78, 80, 73, 82, 75, 68, 85, 87, 89, 89, 87, 87, 95, 97, 97, 97, 101, 102, 104, 106
Offset: 1

Views

Author

Wesley Ivan Hurt, May 03 2020

Keywords

Examples

			a(1) = 5; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its largest side length is 5.
a(6) = 15; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose largest side lengths are 15 and 12. The largest of these is 15.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330917, A330923.

A306991 Number of primitive Heronian isosceles triangles with perimeter A308222(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 2, 3, 3, 5, 4, 3, 6, 4, 4, 5, 4, 8, 6, 9, 6, 6, 4, 8, 11, 8, 10, 6, 9, 9, 8, 14, 6, 15, 10, 11, 10, 8, 12, 10, 12, 18, 9, 12, 12, 20, 14, 8, 21, 15, 12, 16, 10, 23, 16, 21, 12, 12, 16, 18, 26, 18, 20, 12, 16, 18, 20, 29, 12, 21, 30
Offset: 1

Views

Author

Peter Kagey, May 18 2019

Keywords

Comments

Records occur at indices of 1, 4, 6, 11, 14, 19, 21, 26, 33, 35, ... corresponding to perimeters of 16, 50, 98, 242, 338, 578, 722, 1058, 1682, 1922, ...

Examples

			For n = 6, the a(6) = 3 primitive Heronian isosceles triangles with perimeter A308222(6) = 98 have side lengths (24,37,37), (25,25,48), and (29,29,40), and areas of 420, 168, and 420 respectively.

Crossrefs

Cf. A298614, A305717, A308222.

A330922 Largest possible area of a Heronian triangle with perimeter A051518(n).

Original entry on oeis.org

6, 12, 12, 24, 30, 48, 60, 60, 84, 66, 108, 120, 126, 84, 150, 192, 132, 204, 210, 240, 114, 156, 300, 336, 264, 360, 432, 420, 480, 468, 540, 330, 588, 456, 600, 756, 768, 780, 726, 816, 840, 972, 1080, 456, 924, 1170, 1200, 1260, 984, 1344, 1020, 1290, 522, 1452
Offset: 1

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			a(1) = 6; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its area is 3*4/2 = 6.
a(6) = 48; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], with areas 24 and 48. The largest area of the two triangles is 48.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Showing 1-10 of 10 results.

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