A019489 Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(3,7).

3, 7, 16, 36, 80, 177, 391, 863, 1904, 4200, 9264, 20433, 45067, 99399, 219232, 483532, 1066464, 2352161, 5187855, 11442175, 25236512, 55660880, 122763936, 270764385, 597189651, 1317143239, 2905050864, 6407291380, 14131726000, 31168502865, 68744297111

Offset: 0

Author

R. K. Guy

Keywords

• nonn

Comments

a(n) = A077852(n+1) (Barker's recurrence) is correct at least up to n=32000. - R. J. Mathar, Feb 11 2016

Not to be confused with the Pisot T(3,7) sequence, which is A020746. - R. J. Mathar, Feb 13 2016

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
• Index entries for Pisot sequences

Programs

Maple

A019489 := proc(n)
        option remember;
        if n <= 1 then
                op(n+1,[3,7]) ;
        else
                a := procname(n-1)^2/procname(n-2) ;
                if type(a,'integer') then
                        a-1 ;
                else
                        floor(a) ;
                fi;
        end if;
end proc: # R. J. Mathar, Feb 11 2016

PARI

T(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=ceil(a[n-1]^2/a[n-2])-1); a
T(3, 7, 30) \\ Colin Barker, Feb 16 2016

Formula

Empirical G.f.: -(x^3-x^2+2*x-3) / ((x-1)*(x^3+2*x-1)). [Colin Barker, Dec 21 2012]

a(n+1) = ceiling(a(n)^2/a(n-1))-1 for n>0. - Bruno Berselli, Feb 15 2016