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

1, 2, 4, 8, 9, 14, 16, 17, 19, 21, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105

Offset: 1

Author

Clark Kimberling, Nov 26 2010

Keywords

• nonn

Comments

Let S be the set generated by these rules:

(1) if m and n are in S and m

(2) one or more specific numbers are in S by decree.

The floor-sum sequence determined by (1) and (2) results by arranging the elements of S in strictly increasing order.

Examples

			Viewing the floor-sum as a binary operation o, we create S in successive generations:
1, 2 (0th generation);
1o2=4 (1st generation);
1o4=8, 2o4=9 (2nd generation);
1o8=14, 2o8=16, 4o8=19 and four others (3rd generation).
		

Links

• Iain Fox, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A182654, A182655, A182656.

Programs

Maple

A182653 := proc(amax)
        a := {1,2} ;r := (1+sqrt(5))/2 ;
        while true do
                anew := {} ;
                for i in a do
                for j in a do
                        if i <> j then
                                S := floor(r*(i+j)) ;
                                if is(S <= amax) then
                                        anew := anew union { S };
                                end if;
                        end if;
                end do:
                end do:
                if a union anew = a then
                        return sort(a) ;
                end if;
                a := a union anew ;
        end do:
end proc:
A182653(106) ;

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

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