cp's OEIS Frontend

A007066 a(n) = 1 + ceiling((n-1)*phi^2), phi = (1+sqrt(5))/2.

%I A007066 M3299 #69 Aug 04 2025 18:51:36
%S A007066 1,4,7,9,12,15,17,20,22,25,28,30,33,36,38,41,43,46,49,51,54,56,59,62,
%T A007066 64,67,70,72,75,77,80,83,85,88,91,93,96,98,101,104,106,109,111,114,
%U A007066 117,119,122,125,127,130,132,135,138,140,143,145,148,151,153,156,159,161,164,166
%N A007066 a(n) = 1 + ceiling((n-1)*phi^2), phi = (1+sqrt(5))/2.
%C A007066 First column of dual Wythoff array, A126714.
%C A007066 Positions of 0's in A189479.
%C A007066 Skala (2016) asks if this sequence also gives the positions of the 0's in A283310. - _N. J. A. Sloane_, Mar 06 2017
%C A007066 Upper Wythoff sequence plus 2, when shifted by 1. - _Michel Dekking_, Aug 26 2019
%C A007066 In the Fokkink-Joshi paper, this sequence is the Cloitre (0,1,2,3)-hiccup sequence, i.e., a(1) = 1; for m < n, a(n) = a(n-1)+2 if a(m) = n, else a(n) = a(n-1)+3. - _Michael De Vlieger_, Jul 30 2025
%D A007066 Clark Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
%D A007066 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.
%D A007066 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007066 Reinhard Zumkeller, <a href="/A007066/b007066.txt">Table of n, a(n) for n = 1..10000</a>
%H A007066 Benoit Cloitre, <a href="https://arxiv.org/abs/2506.18103">A study of a family of self-referential sequences</a>, arXiv:2506.18103 [math.GM], 2025. See p. 9.
%H A007066 Robbert Fokkink and Gandhar Joshi, <a href="https://arxiv.org/abs/2507.16956">On Cloitre's hiccup sequences</a>, arXiv:2507.16956 [math.CO], 2025. See pp. 3-4, 7, 10.
%H A007066 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions</a>
%H A007066 Matthew Skala, <a href="https://arxiv.org/abs/1604.04072">Graph Nimors</a>, arXiv preprint arXiv:1604.04072 [math.CO], 2016.
%H A007066 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>
%F A007066 a(n) = floor(1+phi*floor(phi*(n-1)+1)), phi=(1+sqrt(5))/2, n >= 2.
%F A007066 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
%F A007066 a(n+1) = floor(n*phi^2) + 2, n>=1. - _Michel Dekking_, Aug 26 2019
%p A007066 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;
%t A007066 t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0, 1}}] &, {0}, 6] (*A189479*)
%t A007066 Flatten[Position[t, 0]] (*A007066*)
%t A007066 Flatten[Position[t, 1]] (*A099267*)
%t A007066 With[{grs=GoldenRatio^2},Table[1+Ceiling[grs(n-1)],{n,70}]] (* _Harvey P. Dale_, Jun 24 2011 *)
%o A007066 (Haskell)
%o A007066 a007066 n = a007066_list !! (n-1)
%o A007066 a007066_list = 1 : f 2 [1] where
%o A007066    f x zs@(z:_) = y : f (x + 1) (y : zs) where
%o A007066      y = if x `elem` zs then z + 2 else z + 3
%o A007066 -- _Reinhard Zumkeller_, Sep 26 2014, Sep 18 2011
%o A007066 (Python)
%o A007066 from math import isqrt
%o A007066 def A007066(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n if n > 1 else 1 # _Chai Wah Wu_, Aug 25 2022
%Y A007066 Cf. A064437.
%Y A007066 Apart from initial terms, same as A026356 (Cloitre (0,2,2,3)-hiccup sequence).
%Y A007066 First column of A126714.
%Y A007066 Complement is (essentially) A026355.
%Y A007066 Equals 1 + A004957, also n + A004956.
%Y A007066 First differences give A076662.
%Y A007066 Complement of A099267. [_Gerald Hillier_, Dec 19 2008]
%Y A007066 Cf. A193214 (primes). Except for the first term equal to A001950 + 2.
%Y A007066 Cf. A026352 (Cloitre (1,1,2,3)-hiccup sequence), A064437 (Cloitre (0,1,3,2)-hiccup sequence).
%K A007066 nonn,easy,nice
%O A007066 1,2
%A A007066 _N. J. A. Sloane_, _Mira Bernstein_