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A081834 a(1)=1, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+3 otherwise.

%I A081834 #20 Jul 29 2025 15:57:05
%S A081834 1,4,7,11,14,17,21,24,27,30,34,37,40,44,47,50,54,57,60,63,67,70,73,77,
%T A081834 80,83,87,90,93,97,100,103,106,110,113,116,120,123,126,130,133,136,
%U A081834 139,143,146,149,153,156,159,163,166,169,172,176,179,182,186,189,192,196
%N A081834 a(1)=1, a(n)=a(n-1)+4 if n is already in the sequence, a(n)=a(n-1)+3 otherwise.
%C A081834 In the Fokkink-Joshi paper, this sequence is the Cloitre (0,1,4,3)-hiccup sequence, - _Michael De Vlieger_, Jul 29 2025
%H A081834 Reinhard Zumkeller, <a href="/A081834/b081834.txt">Table of n, a(n) for n = 1..10000</a>
%H A081834 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. 7.
%H A081834 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 p. 3.
%F A081834 a(n) = floor(rn-(3r-1)/(r+1)) where r = (1/2) *(3+sqrt(13)).
%t A081834 Block[{c, k, nn}, nn = 10^4; c[_] := False; k = 1; c[1] = True; {k}~Join~Reap[Do[If[c[n], k += 4, k += 3]; c[k] = True; Sow[k], {n, 2, nn}] ][[-1, 1]] ] (* _Michael De Vlieger_, Jul 02 2025 *)
%o A081834 (Haskell)
%o A081834 a081834 n = a081834_list !! (n-1)
%o A081834 a081834_list = 1 : f 2 [1] where
%o A081834    f x zs@(z:_) = y : f (x + 1) (y : zs) where
%o A081834      y = z + (if x `elem` zs then 4 else 3)
%o A081834 -- _Reinhard Zumkeller_, Sep 26 2014
%Y A081834 Cf. A064437, A080590, A080600, A080667, A081842.
%K A081834 nonn
%O A081834 1,2
%A A081834 _Benoit Cloitre_, Apr 11 2003