A074074 -id:A074074 - cp's OEIS frontend

A074075 Values of m corresponding to the minimization problem of A074074.

1, 1, 2, 4, 2, 4, 3, 8, 8, 3, 7, 11, 8, 14, 6, 15, 15, 11, 19, 8, 13, 5, 14, 5, 25, 26, 20, 27, 26, 22, 19, 31, 26, 33, 35, 22, 29, 23, 7, 31, 25, 32, 37, 17, 43, 28, 47, 29, 49, 23, 31, 45, 32, 21, 49, 41, 47, 35, 43, 50, 37, 53, 38, 59, 62, 64, 26, 62, 58, 67, 43, 70, 44, 73, 74, 73, 20

Offset: 1

Lekraj Beedassy, Aug 28 2002

nonn

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

Cf. A074074.

A074074[n_] := Module[{dd, sols, x, y}, dd = Table[(2n+1)^2 - 4 m^2, {m, 1, n}]; sols = Table[{d, x /. Solve[x > 0 && y > 0 && x^2 - d y^2 == 1, {x, y}, Integers]}, {d, dd}] /. C[1] -> 1 // Select[#, #[[2]] != {}&]&; MinimalBy[sols, #[[2, 1]]&][[1, 1]]];
a[n_] := a[n] = Sqrt[(2n+1)^2 - A074074[n]]/2;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 27 2024 *)

a(n) = sqrt( (2n+1)^2-A074074(n))/2 .

Sequence extended beyond a(7) by R. J. Mathar, Sep 21 2009

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.

Original entry on oeis.org

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

Lekraj Beedassy, Aug 28 2002

nonn

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.

Ray Chandler, Table of n, a(n) for n = 1..499
Eric Weisstein's World of Mathematics, Heronian Triangle.
Wikipedia, Heronian Triangle.

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

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 xA074076(n),n=1..80) ; # R. J. Mathar, Sep 21 2009

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.

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

cp's OEIS Frontend

A074075 Values of m corresponding to the minimization problem of A074074.

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