A026366 -id:A026366 - cp's OEIS frontend

A026367 a(n) = least k such that s(k) = n, where s = A026366.

1, 2, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 53, 54, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 75, 76, 77, 78, 80, 81, 83, 84, 86

Offset: 1

Clark Kimberling

nonn

Complement of A026368; also the rank transform (as at A187224) of A004526 (after removal of the initial two zeros).  - Clark Kimberling, Mar 10 2011
Gives the positions of the 1's in A285431. - Jeffrey Shallit, Oct 21 2023
Conjecture:  -1 < n*r - a(n) < 2 for n>=1, where r = (1 + sqrt(3))/2.  - Clark Kimberling, Apr 29 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000
Robbert Fokkink, Gerard Francis Ortega, and Dan Rust, Corner the Empress, arXiv:2204.11805 [math.CO], 2022. See Table 4.
Jeffrey Shallit, Proof of Irvine's conjecture via mechanized guessing, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.

Cf. A026368, A187224, A004526, A285431.

seqA = Table[Ceiling[n/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]]                                           (* A026367 *)
Complement[Range[Length[seqA]], limseqU]  (* A026368 *)
(* Peter J. C. Moses, Mar 10 2011 *)
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 1, 0}}] &, {0}, 13] (* A285431 *)
Flatten[Position[s, 0]]  (* A026368 *)
Flatten[Position[s, 1]]  (* A026367 *)
(* Clark Kimberling, Apr 28 2017 *)

a(n) = a(n-1)+2 if n is in the set 2A+1, a(n) = a(n-1)+1 if n is not in 2A+1 where 2A+1 = {2a(1)+1,2a(2)+1,2a(3)+1,...} = {3,5,9,11,15,17,19,21,25,27,31,33,...}. a(n) = (1+sqrt(3))/2*n+O(1) For example, a(3) = a(2)+2 = 2+2 = 4 since 3 is in 2A+1. a(10) = a(9)+1 = 12+1 = 13 since 10 is not in 2A+1. - Benoit Cloitre, Apr 23 2008

Complement sequence in first comment corrected by Nathan Fox, Mar 21 2014

A026368 a(n) = greatest k such that s(k) = n, where s = A026366.

Original entry on oeis.org

3, 6, 11, 14, 19, 22, 25, 28, 33, 36, 41, 44, 47, 50, 55, 58, 63, 66, 71, 74, 79, 82, 85, 88, 93, 96, 101, 104, 107, 110, 115, 118, 123, 126, 131, 134, 139, 142, 145, 148, 153, 156, 161, 164, 167, 170, 175, 178, 183, 186, 189, 192, 197

Offset: 1

Clark Kimberling

nonn

Appears to be complement of A026367. - N. J. A. Sloane, Oct 18 2022
Complement of the rank transform of the sequence A004526=(1,1,2,2,3,3,4,4,5,5,...). See A187224.
Positions of 0 in the fixed point of the morphism 0->11, 1->110; see A285431. Conjecture: -2 < n*r - a(n) < 4 for n>=1, where r = 2 + sqrt(3). - Clark Kimberling, Apr 29 2017
Also, with an initial 0, appears to be the sequence B' of P-positions in Fraenkel's (2,1)-Wythoff's game. The associated A' sequence is A026367. - N. J. A. Sloane, Oct 20 2022

Clark Kimberling, Table of n, a(n) for n = 1..10000
Robbert Fokkink, Gerard Francis Ortega, and Dan Rust, Corner the Empress, arXiv:2204.11805 [math.CO], 2022. See Table 4.
Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, Aug 27 2014. See Table 4.
J. Shallit, Proof of Irvine's conjecture via mechanized guessing, arXiv preprint arXiv:2310.14252 [math.CO], October 22 2023.

Cf. A026367, A187224, A004526, A285421.

s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 1, 0}}] &, {0}, 13] (* A285431 *)
Flatten[Position[s, 0]]  (* A026368 *)
Flatten[Position[s, 1]]  (* A026367 *)
(* Clark Kimberling, Apr 28 2017 *)

A026369 a(n) = sum of the numbers between the two n's in A026366.

Original entry on oeis.org

2, 8, 32, 50, 97, 127, 161, 199, 285, 335, 444, 506, 572, 642, 790, 872, 1043, 1137, 1331, 1437, 1654, 1772, 1894, 2020, 2276, 2414, 2693, 2843, 2997, 3155, 3473, 3643, 3984, 4166, 4530, 4724, 5111, 5317, 5527, 5741, 6167

Offset: 1

Clark Kimberling

nonn

A045671 Extension of Beatty sequence; complement of A045672.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70

Offset: 0

Aviezri S. Fraenkel

nonn

Sequence can also be characterized by a special numeration system-see above reference.
(s,t)-sequences; the case s=2, t=2.
For n>=1, these are the positions of 1 in the fixed point of the morphism 0->11, 1->1110; see A285671.  Conjecture: -1 < n*r - a(n) < 2 for n>=0, where r = (1 + sqrt(17))/4. - Clark Kimberling, May 02 2017

Shiri Artstein-avidan, Aviezri S. Fraenkel and Vera T. Sós, A two-parameter family of an extension of Beatty sequences, Discrete Math., 308 (2008), 4578-4588. doi:10.1016/j.disc.2007.08.070
A. S. Fraenkel, Heap games, numeration systems and sequences, Annals of Combinatorics, 2 (1998), 197-210.
A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Index entries for sequences related to Beatty sequences

Cf. A026366, A045672, A285671.

s=2; t=2;
mex:=First[Complement[Range[1,Max[#1]+1],#1]]&;
a[0]=0; b[n_]:=b[n]=s*a[n]+t*n;
a[n_]:=a[n]=mex[Flatten[Table[{a[i],b[i]},{i,0,n-1}]]];
Table[a[n],{n,200}] (* A045671 *)
Table[b[n],{n,200}] (* A045672 *)
(* Clark Kimberling, Apr 02 2011 *)
s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 1, 1, 0}}] &, {0}, 10]; (* A285671 *)
Flatten[Position[s, 0]];  (* A045672 *)
Flatten[Position[s, 1]];  (* A045671 *)
(* - Clark Kimberling, May 02 2017 *)

a(n) = mex{a(i), b(i):0 <= iA045672, mex S=least integer >= 0 not in sequence S.

a(n) = (1+sqrt(17))/4*n+O(1). - Benoit Cloitre, Apr 23 2008

cp's OEIS Frontend

A026367 a(n) = least k such that s(k) = n, where s = A026366.

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A026368 a(n) = greatest k such that s(k) = n, where s = A026366.

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A026369 a(n) = sum of the numbers between the two n's in A026366.

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A045671 Extension of Beatty sequence; complement of A045672.

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