A134587 -id:A134587 - cp's OEIS frontend

A239246 Number of primitive Heronian triangles with n as greatest side length.

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

Offset: 1

Frank M Jackson, Mar 13 2014

nonn

			a(17)=3 as there are 3 primitive Heronian triangles with greatest side length of 17. They are (9, 10, 17), (8, 15, 17) and (16, 17, 17).

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heronian Triangle.

Cf. A054875, A070138, A096467, A123323, A134587.

nn=200; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s]&&GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0&&IntegerQ[Sqrt[area2]], AppendTo[lst, c]]], {c, 3, nn}, {b, c}, {a, b}]; Table[Length@Select[lst, #==n &], {n, 1, nn}] (* using T. D. Noe's program at A083875 *)

A134588 A modified Heron sequence starting from 1, 2.

Original entry on oeis.org

1, 2, 3, 10, 27, 98, 120, 327, 196745, 277248, 312987, 405769, 456300, 575532, 702219

Offset: 1

Shanzhen Gao, Nov 02 2007

A modified Heron sequence is an increasing sequence such that every three consecutive terms (say u, v, w) of which determine a Heron triangle by using u+v, u+w and v+w as three sizes. A Heron triangle is a triangle with integer sides and integer area.

Given u

Paul Yiu, K. R. S. Sastry and Shanzhen Gao, Heron Sequences, presented on the 2007 Integers Conference and INTEGERS - Electronic Journal of Combinatorial Number Theory, Article 16 Volume 9 Supplement (2009).

Cf. A134587.

a(9)-a(15) from Rémy Sigrist, Jun 03 2019

A134589 A modified Heron sequence starting from 1,3.

Original entry on oeis.org

1, 3, 12, 49, 108, 243, 624, 1681, 5616, 47628, 49300, 27361200, 29949750, 920048500, 1054364310, 1278524208, 9992125972

Offset: 1

Shanzhen Gao, Nov 02 2007

Paul Yiu, K. R. S. Sastry and Shanzhen Gao, Heron Sequences, presented on the 2007 Integers Conference and INTEGERS - Electronic Journal of Combinatorial Number Theory, Article 16 Volume 9 Supplement (2009).

Cf. A134587, A134588.

a(8)-a(17) from Rémy Sigrist, Jun 03 2019

A134590 A modified Heron sequence starting from 2,5.

Original entry on oeis.org

2, 5, 63, 112, 140, 315, 364, 1261, 3025260, 3730654144, 5513566500, 28558451740

Offset: 1

Shanzhen Gao, Nov 02 2007

Paul Yiu, K. R. S. Sastry, Shanzhen Gao, Heron sequences and their modifications.
Paul Yiu, K. R. S. Sastry and Shanzhen Gao, Heron Sequences, presented on the 2007 Integers Conference and INTEGERS - Electronic Journal of Combinatorial Number Theory, Article 16 Volume 9 Supplement (2009).

Cf. A134587, A134588, A134589.

a(8)-a(12) from Rémy Sigrist, Jun 03 2019

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A239246 Number of primitive Heronian triangles with n as greatest side length.

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A134588 A modified Heron sequence starting from 1, 2.

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A134589 A modified Heron sequence starting from 1,3.

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A134590 A modified Heron sequence starting from 2,5.

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