A271489 Maximal term of TRIP-Stern sequence of level n corresponding to permutation triple (e,132,e).

1, 2, 3, 4, 5, 7, 10, 13, 18, 25, 34, 46, 64, 85, 117, 163, 217, 298, 415, 553, 759, 1057, 1408, 1933, 2692, 3586, 4923, 6856, 9133, 12538, 17461, 23260, 31932, 44470, 59239, 81325, 113257, 150871, 207120, 288445, 384241

Offset: 0

Author

N. J. A. Sloane, Apr 13 2016

Keywords

• nonn
• more

Links

• I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015-2017. See Conjecture 5.8.

Crossrefs

For sequences mentioned in Conjecture 5.8 of Amburg et al. (2015) see A271485, A000930, A271486, A271487, A271488, A164001, A000045, A271489.

Programs

Maple

A271489T := proc(n)
    option remember;
    local an,nrecur ;
    if n = 1 then
        [1,1,1] ;
    else
        an := procname(floor(n/2)) ;
        if type(n,'even') then
            # apply F0
            [op(3,an),op(1,an)+op(3,an),op(2,an)] ;
        else
            # apply F1
            [op(1,an),op(2,an),op(1,an)+op(3,an)] ;
        end if;
    end if;
end proc;
A271489 := proc(n)
    local a,l,nmax;
    a := 0 ;
    for l from 2^n to 2^(n+1)-1 do
        nmax := max( op(A271489T(l)) );
        a := max(a,nmax) ;
    end do:
    a ;
end proc: # R. J. Mathar, Apr 16 2016

Mathematica

A271487T[n_] := A271487T[n] = Module[{an}, If[n == 1, {1, 1, 1}, an = A271487T[Floor[n/2]]; If[EvenQ[n], {an[[3]], an[[1]] + an[[3]], an[[2]]}, {an[[1]], an[[2]], an[[1]] + an[[3]]}]]];
a[n_] := a[n] = Module[{a = 0, l, nMax}, For[l = 2^n, l <= 2^(n + 1) - 1, l++, nMax = Max[A271487T[l]]; a = Max[a, nMax]]; a];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 17 2017, after R. J. Mathar *)

Formula

Conjectures from Lars Blomberg, Jan 08 2018: (Start)

n mod 3 == 0: a(n)=a(n-1)+a(n-4) for n>5.

n mod 3 == 1: a(n)=a(n-1)+a(n-4)-a(n-10) for n>9.

n mod 3 == 2: a(n)=a(n-1)+a(n-4)-a(n-14)-a(n-21) for n>22.

(End)

Conjectures from Colin Barker, Jan 09 2018: (Start)

G.f.: (1 + 2*x + 3*x^2 + 2*x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^10 - x^13) / (1 - 2*x^3 - x^6 - x^9).

a(n) = 2*a(n-3) + a(n-6) + a(n-9) for n>8.

(End)

Extensions

a(11)-a(20) b R. J. Mathar, Apr 16 2016

a(21)-a(40) from Lars Blomberg, Jan 08 2018