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A182653 Floor-sum sequence of r with r = golden ratio = (1+sqrt(5))/2 and a(1)=1, a(2)=2.

%I A182653 #17 Apr 25 2019 03:19:18
%S A182653 1,2,4,8,9,14,16,17,19,21,24,25,27,29,30,32,33,35,37,38,40,42,43,45,
%T A182653 46,48,50,51,53,55,56,58,59,61,63,64,66,67,69,71,72,74,76,77,79,80,82,
%U A182653 84,85,87,88,90,92,93,95,97,98,100,101,103,105
%N A182653 Floor-sum sequence of r with r = golden ratio = (1+sqrt(5))/2 and a(1)=1, a(2)=2.
%C A182653 Let S be the set generated by these rules:
%C A182653 (1) if m and n are in S and m<n, then floor(mr+nr) is in S;
%C A182653 (2) one or more specific numbers are in S by decree.
%C A182653 The floor-sum sequence determined by (1) and (2) results by arranging the elements of S in strictly increasing order.
%H A182653 Iain Fox, <a href="/A182653/b182653.txt">Table of n, a(n) for n = 1..3000</a>
%e A182653 Viewing the floor-sum as a binary operation o, we create S in successive generations:
%e A182653 1, 2 (0th generation);
%e A182653 1o2=4 (1st generation);
%e A182653 1o4=8, 2o4=9 (2nd generation);
%e A182653 1o8=14, 2o8=16, 4o8=19 and four others (3rd generation).
%p A182653 A182653 := proc(amax)
%p A182653         a := {1,2} ;r := (1+sqrt(5))/2 ;
%p A182653         while true do
%p A182653                 anew := {} ;
%p A182653                 for i in a do
%p A182653                 for j in a do
%p A182653                         if i <> j then
%p A182653                                 S := floor(r*(i+j)) ;
%p A182653                                 if is(S <= amax) then
%p A182653                                         anew := anew union { S };
%p A182653                                 end if;
%p A182653                         end if;
%p A182653                 end do:
%p A182653                 end do:
%p A182653                 if a union anew = a then
%p A182653                         return sort(a) ;
%p A182653                 end if;
%p A182653                 a := a union anew ;
%p A182653         end do:
%p A182653 end proc:
%p A182653 A182653(106) ;
%o A182653 (PARI) lista(nn) = my(S=[1, 2], r=(1+sqrt(5))/2, new, k); while(1, new=[]; for(m=1, #S, for(n=m+1, #S, k=floor(r*(S[m]+S[n])); if(k<=nn, new=setunion(new,[k])))); if(S==setunion(S,new), return(S)); S=setunion(S,new)) \\ _Iain Fox_, Apr 24 2019
%Y A182653 Cf. A182654, A182655, A182656.
%K A182653 nonn
%O A182653 1,2
%A A182653 _Clark Kimberling_, Nov 26 2010