A026363 a(n) is the least k such that s(k) = n, where s = A026362.

1, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 79, 80, 82, 83, 85, 86

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Or, starting from the natural number, delete successively from the working sequence the term in position 2*a(n). From natural numbers, delete the term in position 2*1, i.e., 2. This leaves 1,3,4,5,6,7,8,9,10,11,... . Delete now the term in position 2*3=6, i.e., 7. This leaves 1,3,4,5,6,8,9,10,11,... . Delete now the term in position 2*4=8, i.e., 10. This leaves 1,3,4,5,6,8,9,11,... and so on. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007

The term deleted from the n-th working sequence is equal to A026364(n), which means that all integers which are not in the present sequence are in A026364 and no others. - Philippe Lallouet (philip.lallouet(AT)orange.fr), May 05 2008

Complement of A026364; also the rank transform (as at A187224) of A004526 after removal of its first three terms, leaving (1,2,2,3,3,4,4,5,5,6,6,...). - Clark Kimberling, Mar 10 2011

Positions of 1 in the fixed point of the morphism 0->11, 1->101; see A285430.

Conjecture: -1 < n*r - a(n) < 2 for n>=1, where r = (1 + sqrt(3))/2. - Clark Kimberling, Apr 29 2017

Links

• Carmine Suriano, Table of n, a(n) for n = 1..10000
• Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 5, 12.

Crossrefs

Cf. A026362, A079255, A080428, A187224, A026364, A004526, A285430.

Programs

Mathematica

seqA = Table[Floor[(n+2)/2], {n, 1, 180}] (* A004526 *)
seqB = Table[n, {n, 1, 80}];              (* A000027 *)
jointRank[{seqA_, seqB_}] := {Flatten@Position[#1, {_, 1}],
Flatten@Position[#1, {_, 2}]} &[Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU = FixedPoint[jointRank[{seqA, #1[[1]]}] &, jointRank[{seqA, seqB}]][[1]]                                  (* A026363 *)
Complement[Range[Length[seqA]], limseqU] (* A026364 *)
(* Peter J. C. Moses, Mar 10 2011 *)
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 0, 1}}] &, {0}, 13] (* A285430 *)
Flatten[Position[s, 0]]  (* A026364 *)
Flatten[Position[s, 1]]  (* A026363 *)
(* Clark Kimberling, Apr 28 2017 *)

Formula

a(1)=1, then a(n)=a(n-1)+2 if n is even and n/2 is not in the sequence, a(n)=a(n-1)+1 otherwise (in particular a(2k+1)=a(2k)+1). a(n)=(1+sqrt(3))/2*n+O(1). Taking a(0)=0, for n>=1 a(2n)-a(2n-2)=A080428(n). - Benoit Cloitre, Apr 23 2008