A210696 Triangulations of the disk, G_{1,n}.

1, 2, 5, 16, 48, 164, 559, 1952, 6872, 24520, 88006, 318444, 1158944, 4241688, 15598973, 57620596, 213680472, 795270644, 2969483214, 11121038100, 41763779054, 157235683780, 593355907790, 2243975358216, 8503404201874, 32283434698908, 122779218918272, 467713035691608

Offset: 0

Author

R. J. Mathar, Mar 30 2012

Keywords

• nonn

Comments

This corrects a typographical error in A005497(6).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Jean-François Alcover, Mathematica program
• William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
• William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]

Crossrefs

Row n=1 of A262586.

Programs

Maple

BrownE := proc(r,n,m)
    local j,s,p ;
    if r < 1 then
        return 0 ;
    elif r = 1 then
        return A146305(n,m) ;
    elif r = 2 then
        j := n mod 2 ; s := floor(n/2) ;
        if type(m,'even') then
            return 0 ;
        end if;
        p := (m+1)/2 ;
        if p > 0 and s >= 0 then
            return 2*(2*p)!*(4*s+2*p+2*j-1)!/p!/(p-1)!/s!/(3*s+2*p+2*j)! ;
        else
            return 0 ;
        end if;
    elif r =3 and (n mod 3) =0 and (m mod 3) = 0 then
        s := n/3 ; p := m/3 ;
        if p >= 0 and s >= 0 then
            return (2*p+1)!*(4*s+2*p)!/p!/p!/s!/(3*s+2*p+1)! ;
        else
            return 0 ;
        end if;
    elif r >= 3 then
        if ((n-1) mod r) =0 and ((m+3) mod r) =0 then
            s := (n-1)/r ; p := (m+3)/r-1 ;
            if p>=0 and s>=0 then
            return (2*p+2)!*(4*s+2*p+1)!/p!/(p+1)!/s!/(3*s+2*p+2)! ;
            else
                return 0 ;
            end if;
        else
            return 0 ;
        end if;
    else
        return 0 ;
    end if;
end proc:
BrownG := proc(n,m)
    add( numtheory[phi](s)* BrownE(s,n,m), s = numtheory[divisors](m+3) ) ;
    %/(m+3) ;
end proc:
A210696 := proc(n)
    BrownG(1,n) ;
end proc:
seq(A210696(n),n=0..25) ;

Mathematica

(* See the link section. *)

Extensions

a(26) onwards from Andrew Howroyd, Nov 23 2024