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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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
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.
LINKS
EXAMPLE
a(3) = floor(2r+3r) = 8.
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) ;
PROG
(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
CROSSREFS
Sequence in context: A264235 A160622 A153699 * A234696 A169949 A261984
KEYWORD
nonn
AUTHOR
Clark Kimberling, Nov 27 2010
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)