cp's OEIS Frontend

A026356 a(n) = floor((n-1)*phi) + n + 1, n > 0, where phi = (1+sqrt(5))/2.

%I A026356 #40 Jul 28 2025 10:54:20
%S A026356 2,4,7,9,12,15,17,20,22,25,28,30,33,36,38,41,43,46,49,51,54,56,59,62,
%T A026356 64,67,70,72,75,77,80,83,85,88,91,93,96,98,101,104,106,109,111,114,
%U A026356 117,119,122,125,127,130,132,135,138,140,143,145
%N A026356 a(n) = floor((n-1)*phi) + n + 1, n > 0, where phi = (1+sqrt(5))/2.
%C A026356 Greatest k such that s(k) = n+1, where s = A026354.
%C A026356 Positions of 1 in A189661.
%C A026356 a(n+1) = A001950(n)-2, the Upper Wythoff sequence shifted by 2. - _Michel Dekking_, Oct 18 2018
%C A026356 In the Fokkink-Joshi paper, this sequence is the Cloitre (0,2,2,3)-hiccup sequence, i.e., a(1) = 2; for m < n, a(n) = a(n-1)+2 if a(m) = n, else a(n) = a(n-1)+3. - _Michael De Vlieger_, Jul 28 2025
%H A026356 Michel Dekking, <a href="/A026356/b026356.txt">Table of n, a(n) for n = 1..1000</a>
%H A026356 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 A026356 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. 7, 10, 17.
%t A026356 (* See A189661. Second program: *)
%t A026356 cloitreH[j_, x_, y_, z_, w_ : 120] := Block[{c, k}, c[_] := False; k = x; c[x] = True; {x}~Join~Reap[Do[If[c[n - j], k += y, k += z]; c[k] = True; Sow[k], {n, 2, w}] ][[-1, 1]] ]; cloitreH[0, 2, 2, 3] (* _Michael De Vlieger_, Jul 28 2025 *)
%o A026356 (PARI) r = (1 + sqrt(5))/2;
%o A026356 a(n) = if(n<1, 1, floor((n - 1)* r) + n + 1);
%o A026356 for(n=1, 100, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 25 2017
%o A026356 (Python)
%o A026356 from sympy import sqrt
%o A026356 import math
%o A026356 r=(1 + sqrt(5))/2
%o A026356 def a(n): return 1 if n<1 else int(math.floor((n - 1)*r)) + n + 1
%o A026356 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Mar 25 2017
%o A026356 (Python)
%o A026356 from math import isqrt
%o A026356 def A026356(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n # _Chai Wah Wu_, Aug 11 2022
%Y A026356 Cf. A000201, A026351, etc. Apart from initial terms, same as A007066. Complement is A189662, closely related to A026355.
%K A026356 nonn
%O A026356 1,1
%A A026356 _Clark Kimberling_
%E A026356 Data corrected by _Michel Dekking_, Oct 18 2018