cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A045749 Extension of Beatty sequence; complement of A045750.

Original entry on oeis.org

0, 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
Offset: 0

Views

Author

Aviezri S. Fraenkel

Keywords

Comments

(s,t)-sequences; the case s=3, t=1.
Complement of A045750. It appears likely that A045749(n)=A187570(n) for all n>=1; the equation has been verified for n up to 500. - Clark Kimberling, Apr 02 2011

Links

Crossrefs

Cf. A045681, A045682, A045749, A187570.

Programs

  • Mathematica
    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 *)

Formula

a(n) = mex {a(i), b(i):0 <= iA045750, mex S=least integer >= 0 not in the sequence S.

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

Views

Author

Clark Kimberling, Mar 11 2011

Keywords

Comments

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.

Crossrefs

Cf. A187224, A187571, A045749, A045750.

Programs

  • Mathematica
    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

Views

Author

Aviezri S. Fraenkel

Keywords

Comments

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

Links

Crossrefs

Cf. A045749, A045750.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Aviezri S. Fraenkel

Keywords

Comments

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

References

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

Links

Crossrefs

Cf. A045749, A045750.

Programs

  • Mathematica
    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 *)

Formula

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

A187571 Complement of A187570.

Original entry on oeis.org

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, 265, 269, 273, 277, 281, 285, 292, 296, 300, 307, 311, 315, 322, 326, 330
Offset: 1

Views

Author

Clark Kimberling, Mar 11 2011

Keywords

Comments

Is this the same as A045750?
It seems likely; see the comment at A187570.

Crossrefs

Cf. A187224, A187570, A045750.

Programs

Showing 1-5 of 5 results.

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