A135510 Least positive number missing from row n of Stern's diatomic array (see A049456 or A002487).

2, 3, 4, 6, 6, 14, 20, 28, 38, 54, 90, 150, 216, 350, 506, 876, 1230, 2034, 3160, 4470, 7764, 12190, 18816, 29952, 43800, 73968, 112602, 182210, 285780, 474558, 729432, 1194078, 1843110, 2990880, 4662450, 7608720, 11801580, 18489120, 29790300

Offset: 1

Author

mc (da-da(AT)lycos.de), Feb 09 2008

Keywords

• nonn

Comments

The old definition was "Least numbers not generated by Eisenstein's algorithm: m=1 n=1, then insert between them m+n, at stage p=1. (E.g. next stage (p=2) of Eisenstein's algorithm would be m, m+m+n, m+n, m+n+n, n). The maximum of these symmetric row elements at stage p is fibonacci(p+2); but how to determine the first number not generated at stage p?"

Links

• Don Reble, C++ program for A135510 and A293160
• G. Eisenstein, Über ein einfaches Mittel zur Auffindung der höheren Reciprocitätsgesetze und der mit ihnen zu verbindenden Ergänzungssätze, Journal für die reine und angewandte Mathematik, Volume 39 (1850), page 351ff.
• M. A. Stern, Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193-220.

Crossrefs

Cf. A049456, A002487, A293160.

Programs

Maple

A049456 := proc(n, k)
    option remember;
    if n =1 then
        if k >= 0 and k <=1 then
            1;
        else
            0 ;
        end if;
    elif type(k, 'even') then
        procname(n-1, k/2) ;
    else
        procname(n-1, (k+1)/2)+procname(n-1, (k-1)/2) ;
    end if;
end proc: # R. J. Mathar, Dec 12 2014
mex := proc(L)
        local k;
        for k from 1 do
                if not k in L then
                        return k;
                end if;
        end do:
end proc:
rho:=n->[seq(A049456(n,k),k=0..2^(n-1))];
[seq(mex(rho(n)),n=1..16)]; # N. J. A. Sloane, Oct 14 2017

Mathematica

(* T is A049456 *)
T[n_, k_] := T[n, k] = If[n == 1, If[k >= 0 && k <= 1, 1, 0], If[EvenQ[k], T[n-1, k/2], T[n-1, (k+1)/2] + T[n-1, (k-1)/2]]];
mex[L_] := Module[{k}, For[k = 1, True, k++, If[FreeQ[L, k], Return[k]]]];
rho[n_] := Table[T[n, k], {k, 0, 2^(n-1)}];
a[n_] := a[n] = mex[rho[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Aug 03 2023, after Maple code *)

Extensions

Entry revised by N. J. A. Sloane, Oct 14 2017

a(29)-a(39) from Don Reble, Oct 16 2016