A074076 - OEIS

%I #16 Jul 30 2017 16:32:58

%S 60,4620,2024,5984,11480,22960,41580,8096,45920,521640,226884,392920,

%T 438944,803320,6725544,207900,37966500,1544620,6846840,2295200,

%U 2785484,9009000,4016600,188375200,3383500,149240,5738000,875124,12013456,8848840

%N 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.

%C 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.

%H Ray Chandler, <a href="/A074076/b074076.txt">Table of n, a(n) for n = 1..499</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Heronian_triangle">Heronian Triangle</a>.

%F 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.

%p 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:

%p 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:

%p seq(A074076(n),n=1..80) ; # _R. J. Mathar_, Sep 21 2009

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

%K nonn

%O 1,1

%A _Lekraj Beedassy_, Aug 28 2002

%E Removed assertion that these are the minimum areas - _R. J. Mathar_, Sep 21 2009