A045749 -id:A045749 - cp's OEIS frontend

A045750 Extension of Beatty sequence, complement of A045749.

0, 4, 8, 12, 19, 23, 27, 34, 38, 42, 49, 53, 57, 61, 65, 69, 76, 80, 84, 91, 95, 99, 106, 110, 114, 118, 122, 126, 133, 137, 141, 148, 152, 156, 163, 167, 171, 175, 179, 183, 190, 194, 198, 205, 209, 213, 220, 224, 228, 235, 239, 243, 250, 254, 258

Offset: 0

Aviezri S. Fraenkel

nonn

(s,t)-sequences; the case s=3, t=1.

Complement of A187749. It appears likely that A045750(n)=A187571(n) for all n>=1; the equation has been verified for n up to 500. - Clark Kimberling, Apr 02 2011

A. S. Fraenkel, Heap games, numeration systems and sequences, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), #07.1.4.
Index entries for sequences related to Beatty sequences

Cf. A045671, A045672, A187571.

s=3; t=1;
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}] (* A045749 *)
Table[b[n], {n, 200}] (* A045750 *)
(* Clark Kimberling, Apr 02 2011 *)

a(n) = 3*A045749(n) + n.

A187570 Rank transform of the sequence ceiling(n/3); complement of A187571.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 92, 93, 94, 96, 97, 98, 100, 101, 102, 103, 104, 105, 107, 108, 109, 111, 112, 113, 115, 116

Offset: 1

Clark Kimberling, Mar 11 2011

nonn

Appears to be a duplicate of A045749. -  R. J. Mathar, Mar 15 2011
The Mathematica programs shown at A187570 and A045749 confirm equality of the first 500 terms. - Clark Kimberling, Apr 02 2011
The sequence of which A187570 is the rank transform is (1,1,1,2,2,2,3,3,3,4,4,4,...), which is (A002264 without the initial three zeros). For a discussion on rank transforms, see A187224.

Cf. A187224, A187571, A045749, A045750.

seqA = Table[Ceiling[n/3], {n, 1, 220}]  (*A002264*)
seqB = Table[n, {n, 1, 220}]; (*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]] (*A187570*)
Complement[Range[Length[seqA]], limseqU]  (*A187571*)
(*by Peter J. C. Moses, Mar 11 2011*)

A045774 Extension of Beatty sequence; complement of A045775.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65

Offset: 0

Aviezri S. Fraenkel

nonn

(s,t)-sequences; the case s=3, t=2.

A. S. Fraenkel, Heap games, numeration systems and sequences, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, 179 (2014), 28-43. See Table 5.
Index entries for sequences related to Beatty sequences

Cf. A045749, A045750.

s=3; 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}] (* A045774 *)
Table[b[n],{n,200}] (* A045775 *)
(* Clark Kimberling, Apr 02 2011 *)

a(n)=mex {a(i), b(i):0 <= iA045775, mex S=least integer >= 0 not in the sequence S.

A045775 Extension of Beatty sequence; complement of A045774.

Original entry on oeis.org

0, 5, 10, 15, 20, 28, 33, 38, 43, 51, 56, 61, 66, 74, 79, 84, 89, 97, 102, 107, 112, 117, 122, 127, 135, 140, 145, 150, 158, 163, 168, 173, 181, 186, 191, 196, 204, 209, 214, 219, 224, 229, 234, 242, 247, 252, 257, 265, 270, 275, 280, 288, 293, 298, 303

Offset: 0

Aviezri S. Fraenkel

nonn

(s,t)-sequences; the case s=3, t=2.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

A. S. Fraenkel, Heap games, numeration systems and sequences, Annals of Combinatorics, 2 (1998), 197-210.
Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, 27 August 2014; see Table 5.
Index entries for sequences related to Beatty sequences

Cf. A045749, A045750.

s=3; 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}] (* A045774 *)
Table[b[n],{n,200}] (* A045775 *)
(* From Clark Kimberling, Apr 02 2011 *)

a(n) = 3*A045774(n)+2*n.

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A045750 Extension of Beatty sequence, complement of A045749.

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A187570 Rank transform of the sequence ceiling(n/3); complement of A187571.

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A045774 Extension of Beatty sequence; complement of A045775.

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A045775 Extension of Beatty sequence; complement of A045774.

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