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

2, 3, 8, 16, 17, 29, 30, 32, 38, 40, 50, 51, 53, 55, 56, 59, 61, 64, 66, 67, 69, 72, 74, 76, 77, 79, 84, 85, 87, 88, 90, 92, 93, 95, 98, 100, 101, 103, 106, 108, 110, 111, 113, 114, 116, 118, 119, 121, 122, 124, 126, 127, 129, 131, 132, 134, 135, 137, 139, 140

Offset: 1

Author

Clark Kimberling, Nov 27 2010

Keywords

• nonn

Comments

Let S be the set generated by these rules: (1) if m and n are in S and m

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.

Examples

			a(3) = floor(2r+3r) = 8.
		

Links

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

Crossrefs

Cf. A000201, A182653, A182669.

Programs

Maple

A182670 := proc(amax)
        a := {2,3} ;
        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:
A182670(140) ;

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

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