A074076 One-sixth of the area of some primitive Heronian triangles with a distance of 2n+1 between the median and altitude points on the longest side.

60, 4620, 2024, 5984, 11480, 22960, 41580, 8096, 45920, 521640, 226884, 392920, 438944, 803320, 6725544, 207900, 37966500, 1544620, 6846840, 2295200, 2785484, 9009000, 4016600, 188375200, 3383500, 149240, 5738000, 875124, 12013456, 8848840

Offset

1,1

Comments

With N=2n+1, such a triangle has sides N*u +/- M, 2*M*u (the latter being cut into M*u +/- N by the corresponding altitude) and inradius M*(N - M)*v. The first entry, in particular, is associated with sequence A023039.

Links

Ray Chandler, Table of n, a(n) for n = 1..499

Eric Weisstein's World of Mathematics, Heronian Triangle.

Wikipedia, Heronian Triangle.

Formula

a(n) = M(n)*D(n)*u(n)*v(n)/6, where (u, v) is the fundamental solution to x^2 - D*y^2 = 1, with M = 2*A074075(n); D = A074074(n) = N^2 - M^2.

Maple

A033313 := proc(Dcap) local c, i, fr, nu, de ; if issqr(Dcap) then -1; else c := numtheory[cfrac](sqrt(Dcap)) ; for i from 1 do try fr := numtheory[nthconver](c, i) ; nu := numer(fr) ; de := denom(fr) ; if nu^2-Dcap*de^2=1 then RETURN(nu) ; fi; catch: RETURN(-1) ; end try; od: fi: end:
A074076 := proc(n) local Dmin, xmin, Dcap ; Dmin := -1; xmin := -1; mmin := -1; ymin := -1; for m from 1 to n do Dcap := (2*n+1+2*m)*(2*n+1-2*m) ; x := A033313(Dcap) ; if xmin = -1 or (x >0 and x<xmin ) then mmin := m ; xmin := x ; ymin := sqrt((xmin^2-1)/Dcap) ; Dmin := Dcap ; fi; od: Mmin := 2*mmin ; Mmin*Dmin*xmin*ymin/6 ; end:
seq(A074076(n), n=1..80) ; # R. J. Mathar, Sep 21 2009

Crossrefs

Cf. A074074, A074075, A023039, A002349, A002350.

Sequence in context: A042741 A166772 A293091 * A084274 A289307 A091032

Adjacent sequences: A074073 A074074 A074075 * A074077 A074078 A074079

Keyword

nonn

Author

Lekraj Beedassy, Aug 28 2002

Extensions

Removed assertion that these are the minimum areas - R. J. Mathar, Sep 21 2009

Status

approved