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A182670 Floor-sum sequence of r, where r = golden ratio = (1+sqrt(5))/2 and a(1)=2, a(2)=3.

%I A182670 #17 Apr 25 2019 03:19:43
%S A182670 2,3,8,16,17,29,30,32,38,40,50,51,53,55,56,59,61,64,66,67,69,72,74,76,
%T A182670 77,79,84,85,87,88,90,92,93,95,98,100,101,103,106,108,110,111,113,114,
%U A182670 116,118,119,121,122,124,126,127,129,131,132,134,135,137,139,140
%N A182670 Floor-sum sequence of r, where r = golden ratio = (1+sqrt(5))/2 and a(1)=2, a(2)=3.
%C A182670 Let S be the set generated by these rules:  (1) if m and n are in S and m<n, then floor(mr+nr) is in S; (2) two or more specific numbers are in S.  The floor-sum sequence determined by (1) and (2) results by arranging the elements of S in strictly increasing order.
%C A182670 Let B be the Beatty sequence of r. Then a floor-sum sequence of r is a subsequence of B if and only if a(1) and a(2) are terms of B.  Thus, A182670 is not a subsequence of the lower Wythoff sequence, A000201.
%H A182670 Iain Fox, <a href="/A182670/b182670.txt">Table of n, a(n) for n = 1..3000</a>
%e A182670 a(3) = floor(2r+3r) = 8.
%p A182670 A182670 := proc(amax)
%p A182670         a := {2,3} ;
%p A182670         r := (1+sqrt(5))/2 ;
%p A182670         while true do
%p A182670                 anew := {} ;
%p A182670                 for i in a do for j in a do
%p A182670                         if i <> j then  S := floor(r*(i+j)) ;                     if is(S <= amax) then anew := anew union { S }; end if;
%p A182670                         end if;
%p A182670                 end do:
%p A182670                 end do:
%p A182670                 if a union anew = a then
%p A182670                         return sort(a) ;
%p A182670                 end if;
%p A182670                 a := a union anew ;
%p A182670         end do:
%p A182670 end proc:
%p A182670 A182670(140) ;
%o A182670 (PARI) lista(nn) = my(S=[2, 3], 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 25 2019
%Y A182670 Cf. A000201, A182653, A182669.
%K A182670 nonn
%O A182670 1,1
%A A182670 _Clark Kimberling_, Nov 27 2010