A276871 -id:A276871 - cp's OEIS frontend

A276885 Sums-complement of the Beatty sequence for 1 + phi.

1, 4, 9, 12, 17, 22, 25, 30, 33, 38, 43, 46, 51, 56, 59, 64, 67, 72, 77, 80, 85, 88, 93, 98, 101, 106, 111, 114, 119, 122, 127, 132, 135, 140, 145, 148, 153, 156, 161, 166, 169, 174, 177, 182, 187, 190, 195, 200, 203, 208, 211, 216, 221, 224, 229, 232, 237

Offset: 1

Clark Kimberling, Oct 01 2016

See A276871 for a definition of sums-complement and guide to related sequences.
This appears to be 1 followed by A089910. - R. J. Mathar, Oct 05 2016
Mathar's conjecture is proved in the paper 'The Frobenius problem for homomorphic embeddings of languages into the integers'. See Example 1 in that paper. - Michel Dekking, Dec 21 2017

			The Beatty sequence for 1 + phi is A001950 = (2,5,7,10,13,15,18,20,23,26,...), with difference sequence s = A005614 + 2 = (3,2,3,3,2,3,2,3,3,...). The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,7,8,10,11,13,14,15,16,18,...), with complement (1,4,9,12,17,22,...).

Jeffrey Shallit, "Synchronized Sequences" in Lecture Notes of Computer science 12847 pp 1-19 2021, see page 16.

Michel Dekking, The Frobenius problem for homomorphic embeddings of languages into the integers, arXiv:1712.03345 [math.CO], 2017.
Index entries for sequences related to Beatty sequences

Cf. A001950, A003849, A276871.

z = 500; r = 1 + GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A001950 *)
t = Differences[b]; (* 2 + A003849 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]; (* A276885 *)

from math import isqrt
def A276885(n): return n+(n-1+isqrt(5*(n-1)**2)&-2) # Chai Wah Wu, May 21 2025

a(n) = 2[(n-1)phi] + n, where phi = (1+sqrt(5))/2 (see Example 1 in the paper 'The Frobenius problem for homomorphic embeddings of languages into the integers'). - Michel Dekking, Dec 21 2017

a(n) = A035336(n-1)+2 for n>1. - Michel Dekking, Dec 21 2017

Name edited and example corrected by Michel Dekking, Oct 30 2016

A276876 Sums-complement of the Beatty sequence for 2e.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 12, 13, 14, 15, 18, 19, 20, 23, 24, 25, 26, 29, 30, 31, 34, 35, 36, 37, 40, 41, 42, 45, 46, 47, 50, 51, 52, 53, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69, 72, 73, 74, 75, 78, 79, 80, 83, 84, 85, 88, 89, 90, 91, 94, 95, 96, 99, 100, 101, 102

Offset: 1

Clark Kimberling, Sep 27 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for 2e is A276853 = (0,5,10,16,21,27,32,...), with difference sequence s = A276860 = (5,5,6,5,6,5,6,5,5,6,5,6,5,6,5,5,...).  The sums s(j)+s(j+1)+...+s(k) include (5,6,7,10,11,16,17,...), with complement (1,2,3,4,7,8,9,12,...).

Index entries for sequences related to Beatty sequences

Cf. A276853, A276860, A276871.

z = 500; r = 2E; b = Table[Floor[k*r], {k, 0, z}]; (* A276853 *)
t = Differences[b]; (* A276860 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276876 *)

A276881 Sums-complement of the Beatty sequence for 1 + sqrt(5).

Original entry on oeis.org

1, 2, 5, 8, 11, 14, 15, 18, 21, 24, 27, 28, 31, 34, 37, 40, 41, 44, 47, 50, 53, 54, 57, 60, 63, 66, 69, 70, 73, 76, 79, 82, 83, 86, 89, 92, 95, 96, 99, 102, 105, 108, 109, 112, 115, 118, 121, 124, 125, 128, 131, 134, 137, 138, 141, 144, 147, 150, 151, 154

Offset: 1

Clark Kimberling, Sep 27 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for 1 + sqrt(5) is A276854 = (0,3,6,9,12,16,19,...), with difference sequence s = A276863 = (3,3,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4,...).  The sums s(j)+s(j+1)+...+s(k) include (3,4,6,7,9,10,12,13,...), with complement (1,2,5,8,11,14,15,,...).

Index entries for sequences related to Beatty sequences

Cf. A276854, A276863, A276871.

z = 500; r = 1+ Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A276854 *)
t = Differences[b]; (* A276863 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276881 *)

A276886 Sums-complement of the Beatty sequence for 2 + phi.

Original entry on oeis.org

1, 2, 5, 6, 9, 12, 13, 16, 17, 20, 23, 24, 27, 30, 31, 34, 35, 38, 41, 42, 45, 46, 49, 52, 53, 56, 59, 60, 63, 64, 67, 70, 71, 74, 77, 78, 81, 82, 85, 88, 89, 92, 93, 96, 99, 100, 103, 106, 107, 110, 111, 114, 117, 118, 121, 122, 125, 128, 129, 132, 135, 136

Offset: 1

Clark Kimberling, Oct 01 2016

See A276871 for a definition of sums-complement and guide to related sequences.
From Michel Dekking, Apr 30 2019: (Start)
This sequence is a generalized Beatty sequence. According to Theorem 3.2 in the paper "The Frobenius problem for homomorphic embeddings of languages into the integers" this sequence (as a subset of the natural numbers) is the complement of the union of the two Beatty sequences
     V := A003231 and W = V+1 (as subsets of the natural numbers) given by
     V(n):= A(n)+2n = 3,7,10,14,..., W(n):=A(n)+2n+1 = 4,8,11,15,...
Here A = A000201, the lower Wythoff sequence.
Since the sequence Delta A = A014675 of first differences of A is the infinite Fibonacci word on the alphabet {2,1}, the sequence Delta V = (V(n+1)-V(n)) is the infinite Fibonacci word on the alphabet {4,3}. (Delta V equals A276867 shifted by 1.)
Now if for some k, Delta V(k) = 4, then a distance 3 plus a distance 1 are generated between three consecutive numbers in the complement, whereas if Delta V(k) = 3, then only a distance 3 is generated between two consecutive numbers in the complement.
This means that (skipping a(1)=1)
    Delta a = (a(n+1)-a(n)) = gamma(Delta V),
where gamma is the morphism
    gamma(4) = 31, gamma(3) = 3.
Since the Fibonacci word is a fixed point of the morphism 0->01, 1->0, this implies that Delta a, skipping a(1)=1, is the Fibonacci word on the alphabet {3,1}. It follows that
    a(n+1) = 2*A(n) - n + 1.
(End)

			The Beatty sequence for 2 + phi is 0 followed by A003231, which is (0,3,7,10,14,18,21,...), with difference sequence s = A276867 = (3,4,3,4,4,3,4,3,4,4,3,4,4,3,4,3,4,4,3,...). The sums s(j)+s(j+1)+...+s(k) include (3,4,7,8,10,12,14,15,...), with complement (1,2,5,6,9,12,13,16,...).

J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
Michel Dekking, The Frobenius problem for homomorphic embeddings of languages into the integers, Theoretical Computer Science 732, 7 July 2018, 73-79.
Index entries for sequences related to Beatty sequences

Cf. A003231, A276867, A276871, A296184(2+phi).

z = 500; r = 2 + GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A003231 *)
t = Differences[b]; (* A276867 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w];  (* A276886 *)

a(n) = 2*floor((n-1)*phi) - n + 2, where phi is the golden mean.

A276888 Sums-complement of the Beatty sequence for 2 + sqrt(1/2).

Original entry on oeis.org

1, 4, 7, 12, 15, 20, 23, 26, 31, 34, 39, 42, 45, 50, 53, 58, 61, 66, 69, 72, 77, 80, 85, 88, 91, 96, 99, 104, 107, 112, 115, 118, 123, 126, 131, 134, 137, 142, 145, 150, 153, 156, 161, 164, 169, 172, 177, 180, 183, 188, 191, 196, 199, 202, 207, 210, 215, 218

Offset: 1

Clark Kimberling, Oct 01 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for 2 + sqrt(1/2) is A182969 = (0,2,5,8,10,13,16,18,21,...), with difference sequence s = A276869 = (2,3,3,2,3,3,2,3,3,3,2,3,3,2,3,3,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,8,9,10,11,13,14,16,...), with complement (1,4,7,12,15,20,23,...).

Index entries for sequences related to Beatty sequences

Cf. A182769, A276869, A276871.

z = 500; r = 2 + Sqrt[1/2]; b = Table[Floor[k*r], {k, 0, z}]; (* A182769 *)
t = Differences[b]; (* A276869 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w];  (* A276888 *)

A276872 Sums-complement of the Beatty sequence for sqrt(6).

Original entry on oeis.org

1, 6, 11, 16, 21, 28, 33, 38, 43, 50, 55, 60, 65, 70, 77, 82, 87, 92, 99, 104, 109, 114, 119, 126, 131, 136, 141, 148, 153, 158, 163, 168, 175, 180, 185, 190, 197, 202, 207, 212, 217, 224, 229, 234, 239, 246, 251, 256, 261, 268, 273, 278, 283, 288, 295, 300

Offset: 1

Clark Kimberling, Sep 26 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for sqrt(6) is A022840 = (0, 2, 4, 7, 9, 12, 14, 17,...), with difference sequence s = A276856 = (2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,7,8,9,10,12,...), with complement (1,6,11,16,21,...).

Index entries for sequences related to Beatty sequences

Cf. A022840, A276856, A276871.

z = 500; r = Sqrt[6]; b = Table[Floor[k*r], {k, 0, z}]; (* A022840 *)
t = Differences[b]; (* A276856 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w];  (* A276872 *)

A276873 Sums-complement of the Beatty sequence for sqrt(7).

Original entry on oeis.org

1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, 46, 49, 54, 57, 62, 65, 70, 73, 78, 81, 86, 91, 94, 99, 102, 107, 110, 115, 118, 123, 128, 131, 136, 139, 144, 147, 152, 155, 160, 163, 168, 173, 176, 181, 184, 189, 192, 197, 200, 205, 208, 213, 218, 221, 226, 229

Offset: 1

Clark Kimberling, Sep 27 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for sqrt(7) is A022841 = (0,2,5,7,10,13,...), with difference sequence s = A276857 = (2,3,2,3,3,2,3,3,2,3,3,2,3,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,6,7,8,10,11,13,...), with complement (1,4,9,12,17,...).

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See p. 16.
Index entries for sequences related to Beatty sequences

Cf. A022841, A276857, A276871.

z = 500; r = Sqrt[7]; b = Table[Floor[k*r], {k, 0, z}]; (* A022841 *)
t = Differences[b]; (* A276857 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276873 *)

A276874 Sums-complement of the Beatty sequence for sqrt(8).

Original entry on oeis.org

1, 4, 7, 10, 13, 18, 21, 24, 27, 30, 35, 38, 41, 44, 47, 52, 55, 58, 61, 64, 69, 72, 75, 78, 81, 86, 89, 92, 95, 100, 103, 106, 109, 112, 117, 120, 123, 126, 129, 134, 137, 140, 143, 146, 151, 154, 157, 160, 163, 168, 171, 174, 177, 180, 185, 188, 191, 194

Offset: 1

Clark Kimberling, Sep 27 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for sqrt(8) is A022842 = (0,2,5,8,11,14,16,...), with difference sequence s = A276858 = (2,3,3,3,3,2,3,3,3,3,3,2,3,3,3,3,3,2,3,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,8,9,11,12, 14,...), with complement (1,4,7,10,13,18,...)

Index entries for sequences related to Beatty sequences

Cf. A022842, A276858, A276871.

z = 500; r = Sqrt[8]; b = Table[Floor[k*r], {k, 0, z}]; (* A022842 *)
t = Differences[b]; (* A276858 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276874 *)

A276875 Sums-complement of the Beatty sequence for e.

Original entry on oeis.org

1, 4, 7, 12, 15, 18, 23, 26, 31, 34, 37, 42, 45, 50, 53, 56, 61, 64, 69, 72, 75, 80, 83, 88, 91, 94, 99, 102, 105, 110, 113, 118, 121, 124, 129, 132, 137, 140, 143, 148, 151, 156, 159, 162, 167, 170, 175, 178, 181, 186, 189, 194, 197, 200, 205, 208, 211, 216

Offset: 1

Clark Kimberling, Sep 27 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for e is A022843 = (0,2,5,8,10,13,16,...), with difference sequence s = A276859 = (2,3,3,2,3,3,3,2,3,3,2,3,3,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,8,9,10,12,13,...), with complement (1,4,7,12,15,...).

Index entries for sequences related to Beatty sequences

Cf. A022843, A276859, A276871.

z = 500; r = E; b = Table[Floor[k*r], {k, 0, z}]; (* A022843 *)
t = Differences[b]; (* A276859 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276875 *)

A276878 Sums-complement of the Beatty sequence for 2*Pi.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 16, 17, 20, 21, 22, 23, 24, 27, 28, 29, 30, 33, 34, 35, 36, 39, 40, 41, 42, 45, 46, 47, 48, 49, 52, 53, 54, 55, 58, 59, 60, 61, 64, 65, 66, 67, 68, 71, 72, 73, 74, 77, 78, 79, 80, 83, 84, 85, 86, 89, 90, 91, 92, 93, 96

Offset: 1

Clark Kimberling, Sep 27 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for 2*Pi is A038130 = (0,6,12,18,25,31,37,...), with difference sequence s = A276861 = (6,6,6,7,6,6,6,7,6,6,7,...).  The sums s(j)+s(j+1)+...+s(k) include (6,7,12,13,...), with complement (1,2,3,4,5,8,9,10,...).

Index entries for sequences related to Beatty sequences

Cf. A038130, A276861, A276871.

z = 500; r = 2*Pi; b = Table[Floor[k*r], {k, 0, z}]; (* A038130 *)
t = Differences[b]; (* A276861 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276878 *)

cp's OEIS Frontend

A276885 Sums-complement of the Beatty sequence for 1 + phi.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A276876 Sums-complement of the Beatty sequence for 2e.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A276881 Sums-complement of the Beatty sequence for 1 + sqrt(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A276886 Sums-complement of the Beatty sequence for 2 + phi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A276888 Sums-complement of the Beatty sequence for 2 + sqrt(1/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A276872 Sums-complement of the Beatty sequence for sqrt(6).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A276873 Sums-complement of the Beatty sequence for sqrt(7).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A276874 Sums-complement of the Beatty sequence for sqrt(8).

Views

Author

Keywords