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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 20 08:12 EST 2017. Contains 294962 sequences.