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A007066
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a(n) = 1 + ceiling((n-1)*phi^2), phi = (1+sqrt(5))/2.
(Formerly M3299)
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17
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1, 4, 7, 9, 12, 15, 17, 20, 22, 25, 28, 30, 33, 36, 38, 41, 43, 46, 49, 51, 54, 56, 59, 62, 64, 67, 70, 72, 75, 77, 80, 83, 85, 88, 91, 93, 96, 98, 101, 104, 106, 109, 111, 114, 117, 119, 122, 125, 127, 130, 132, 135, 138, 140, 143, 145, 148, 151, 153, 156, 159, 161, 164, 166
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OFFSET
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1,2
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COMMENTS
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First column of dual Wythoff array.
Skala (2016) asks if this sequence also gives the positions of the 0's in A283310. - N. J. A. Sloane, Mar 06 2017
Upper Wythoff sequence plus 2, when shifted by 1. - Michel Dekking, Aug 26 2019
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REFERENCES
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Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
D. R. Morrison, "A Stolarsky array of Wythoff pairs," in A Collection of Manuscripts Related to the Fibonacci Sequence. Fibonacci Assoc., Santa Clara, CA, 1980, pp. 134-136.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Matthew Skala, Graph Nimors, arXiv preprint arXiv:1604.04072 [math.CO], 2016.
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FORMULA
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a(n) = floor(1+phi*floor(phi*(n-1)+1)), phi=(1+sqrt(5))/2, n >= 2.
a(1)=1; for n>1, a(n)=a(n-1)+2 if n is already in the sequence, a(n)=a(n-1)+3 otherwise. - Benoit Cloitre, Mar 06 2003
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MAPLE
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Digits := 100: t := (1+sqrt(5))/2; A007066 := proc(n) if n <= 1 then 1 else floor(1+t*floor(t*(n-1)+1)); fi; end;
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MATHEMATICA
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t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0, 1}}] &, {0}, 6] (*A189479*)
Flatten[Position[t, 0]] (*A007066*)
Flatten[Position[t, 1]] (*A099267*)
With[{grs=GoldenRatio^2}, Table[1+Ceiling[grs(n-1)], {n, 70}]] (* Harvey P. Dale, Jun 24 2011 *)
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PROG
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(Haskell)
a007066 n = a007066_list !! (n-1)
a007066_list = 1 : f 2 [1] where
f x zs@(z:_) = y : f (x + 1) (y : zs) where
y = if x `elem` zs then z + 2 else z + 3
(Python)
from math import isqrt
def A007066(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n if n > 1 else 1 # Chai Wah Wu, Aug 25 2022
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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