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

%I A135510 #39 Aug 03 2023 08:36:02
%S A135510 2,3,4,6,6,14,20,28,38,54,90,150,216,350,506,876,1230,2034,3160,4470,
%T A135510 7764,12190,18816,29952,43800,73968,112602,182210,285780,474558,
%U A135510 729432,1194078,1843110,2990880,4662450,7608720,11801580,18489120,29790300
%N A135510 Least positive number missing from row n of Stern's diatomic array (see A049456 or A002487).
%C A135510 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?"
%H A135510 Don Reble, <a href="/A293160/a293160.txt">C++ program for A135510 and A293160</a>
%H A135510 G. Eisenstein, <a href="https://doi.org/10.1515/crll.1850.39.351">Über ein einfaches Mittel zur Auffindung der höheren Reciprocitätsgesetze und der mit ihnen zu verbindenden Ergänzungssätze</a>, Journal für die reine und angewandte Mathematik, Volume 39 (1850), page 351ff.
%H A135510 M. A. Stern, <a href="http://dx.doi.org/10.1515/crll.1858.55.193">Über eine zahlentheoretische Funktion</a>, J. Reine Angew. Math., 55 (1858), 193-220.
%p A135510 A049456 := proc(n, k)
%p A135510     option remember;
%p A135510     if n =1 then
%p A135510         if k >= 0 and k <=1 then
%p A135510             1;
%p A135510         else
%p A135510             0 ;
%p A135510         end if;
%p A135510     elif type(k, 'even') then
%p A135510         procname(n-1, k/2) ;
%p A135510     else
%p A135510         procname(n-1, (k+1)/2)+procname(n-1, (k-1)/2) ;
%p A135510     end if;
%p A135510 end proc: # _R. J. Mathar_, Dec 12 2014
%p A135510 mex := proc(L)
%p A135510         local k;
%p A135510         for k from 1 do
%p A135510                 if not k in L then
%p A135510                         return k;
%p A135510                 end if;
%p A135510         end do:
%p A135510 end proc:
%p A135510 rho:=n->[seq(A049456(n,k),k=0..2^(n-1))];
%p A135510 [seq(mex(rho(n)),n=1..16)]; # _N. J. A. Sloane_, Oct 14 2017
%t A135510 (* T is A049456 *)
%t A135510 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]]];
%t A135510 mex[L_] := Module[{k}, For[k = 1, True, k++, If[FreeQ[L, k], Return[k]]]];
%t A135510 rho[n_] := Table[T[n, k], {k, 0, 2^(n-1)}];
%t A135510 a[n_] := a[n] = mex[rho[n]];
%t A135510 Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* _Jean-François Alcover_, Aug 03 2023, after Maple code *)
%Y A135510 Cf. A049456, A002487, A293160.
%K A135510 nonn
%O A135510 1,1
%A A135510 mc (da-da(AT)lycos.de), Feb 09 2008
%E A135510 Entry revised by _N. J. A. Sloane_, Oct 14 2017
%E A135510 a(29)-a(39) from _Don Reble_, Oct 16 2016