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 1-7 of 7 results.

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

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.

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 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.