Aviezri S. Fraenkel - cp's OEIS frontend

A080241 Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n} for n >= 0, B_0=0, B_1=1 and for n >= 2, B_n = 2B_{n-1}+(-1)^{A_n}. Sequence gives B_n.

0, 1, 3, 7, 13, 27, 55, 109, 219, 437, 875, 1751, 3501, 7003, 14005, 28011, 56021, 112043, 224085, 448171, 896341, 1792683, 3585365, 7170731, 14341463, 28682925, 57365851, 114731701, 229463403, 458926805, 917853611, 1835707221

Offset: 0

Aviezri S. Fraenkel, Mar 12 2003

nonn

The minimal excluded value of set of nonnegative numbers S is mex S = least nonnegative integer not in S.

The sequence A_n is given in A080240.

A. S. Fraenkel, Home Page
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.

Cf. A080240.

More terms from John W. Layman, May 04 2004

A080240 Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n} for n >= 0, B_0=0, B_1=1 and for n >= 2, B_n = 2B_{n-1}+(-1)^{A_n}. Sequence gives A_n.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 0

Aviezri S. Fraenkel, Mar 12 2003

nonn

The minimal excluded value of set of nonnegative numbers S is mex S = least nonnegative integer not in S.

The sequence B_n is given in A080241.

A. S. Fraenkel, Home Page
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.

Cf. A080241.

More terms from Emeric Deutsch, Apr 13 2005

A045682 Extension of Beatty sequence; complement of A045681.

Original entry on oeis.org

0, 5, 10, 15, 20, 27, 32, 37, 42, 49, 54, 59, 64, 71, 76, 81, 86, 93, 98, 103, 108, 113, 118, 125, 130, 135, 140, 147, 152, 157, 162, 169, 174, 179, 184, 191, 196, 201, 206, 211, 216, 223, 228, 233, 238, 245, 250, 255, 260, 267, 272, 277, 282, 289, 294, 299

Offset: 0

Aviezri S. Fraenkel

nonn

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

A. S. Fraenkel, Heap games, numeration systems and sequences, Annals of Combinatorics, 2 (1998), 197-210.
Index entries for sequences related to Beatty sequences

Cf. A045671, A045672.

s=2; t=3;
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}] (* A045681 *)
Table[b[n],{n,200}] (* A045682 *)
(* From Clark Kimberling, Apr 02 2011 *)

a(n) = 2*A045681(n)+3*n.

A045681 Extension of Beatty sequence; complement of A045682.

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, 28, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73

Offset: 0

Aviezri S. Fraenkel

nonn

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

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.
Index entries for sequences related to Beatty sequences

Cf. A045671, A045672.

s=2; t=3;
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}] (* A045681 *)
Table[b[n],{n,200}] (* A045682 *)
(* From Clark Kimberling, Apr 02 2011 *)

a(n)=mex {a(i), b(i):0<=iA045682, 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.

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

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

Aviezri S. Fraenkel

nonn

(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

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. A045681, A045682, A045749, A187570.

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

A045750 Extension of Beatty sequence, complement of A045749.

Original entry on oeis.org

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.

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.

A038150 Array of numbers used in exotic ternary numeration system, read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 11, 16, 21, 7, 14, 29, 42, 55, 9, 19, 37, 76, 110, 144, 10, 24, 50, 97, 199, 288, 377, 12, 27, 63, 131, 254, 521, 754, 987, 13, 32, 71, 165, 343, 665, 1364, 1974, 2584, 15, 35, 84, 186, 432, 898, 1741, 3571, 5168, 6765, 17, 40, 92, 220, 487

Offset: 0

Aviezri S. Fraenkel

			Top left corner of array:
  1,  3,  8, 21,  55, ...
  2,  6, 16, 42, 110, ...
  4, 11, 29, 76, 199, ...
  5, 14, 37, 97, 254, ...

A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
A. S. Fraenkel, Arrays, numeration systems and Frankenstein games, Theoret. Comput. Sci. 282 (2002), 271-284; preprint.

Rows give A001906, A025169, A002878.
Columns give A026351, A047924, A047925.
Main diagonal gives A047923.
Cf. A026351, A035506.

t[n_, 1] := Floor[(n - 1) GoldenRatio] + 1; t[n_, j_] := Floor[ GoldenRatio^2 t[n, j - 1]] + 1; Table[ t[n - m + 1, m], {n, 11}, {m, n}] // Flatten (* Birkas Gyorgy, Apr 15 2011; modified by Robert G. Wilson v, Apr 15 2011 *)

For n >= 0, A_0^n is the least nonnegative integer not in {A_j^n: 0 <= i < n, j >= 0, A_1^n = 2A_0^n + n, A_j^n = 3A_{j-1}^n - A_{j-2}^n (j >= 2).

a(n,k) = F(2k)*n + F(2k+1)*A026351(n). - Charlie Neder, Feb 07 2019

More terms from Naohiro Nomoto, Jun 07 2001

cp's OEIS Frontend

User: Aviezri S. Fraenkel

A080241 Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n} for n >= 0, B_0=0, B_1=1 and for n >= 2, B_n = 2B_{n-1}+(-1)^{A_n}. Sequence gives B_n.

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A080240 Define two sequences by A_n = mex{A_i,B_i : 0 <= i < n} for n >= 0, B_0=0, B_1=1 and for n >= 2, B_n = 2B_{n-1}+(-1)^{A_n}. Sequence gives A_n.

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A045682 Extension of Beatty sequence; complement of A045681.

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A045681 Extension of Beatty sequence; complement of A045682.

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A045775 Extension of Beatty sequence; complement of A045774.

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A045671 Extension of Beatty sequence; complement of A045672.

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A045749 Extension of Beatty sequence; complement of A045750.

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A045750 Extension of Beatty sequence, complement of A045749.

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