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 1-7 of 7 results.

A276871 Sums-complement of the Beatty sequence for sqrt(5).

Original entry on oeis.org

1, 10, 19, 28, 37, 48, 57, 66, 75, 86, 95, 104, 113, 124, 133, 142, 151, 162, 171, 180, 189, 198, 209, 218, 227, 236, 247, 256, 265, 274, 285, 294, 303, 312, 323, 332, 341, 350, 359, 370, 379, 388, 397, 408, 417, 426, 435, 446, 455, 464, 473, 484, 493, 502
Offset: 1

Views

Author

Clark Kimberling, Sep 24 2016

Keywords

Comments

The sums-complement of a sequence s(1), s(2), ... of positive integers is introduced here as the set of numbers c(1), c(2), ... such that no c(n) is a sum s(j)+s(j+1)+...+s(k) for any j and k satisfying 1 <= j <= k. If this set is not empty, the term "sums-complement" also applies to the (possibly finite) sequence of numbers c(n) arranged in increasing order. In particular, the difference sequence D(r) of a Beatty sequence B(r) of an irrational number r > 2 has an infinite sums-complement, abbreviated as SC(r) in the following table:
r B(r) D(r) SC(r)
----------------------------------------------------
sqrt(5) A022839 A081427 A276871
sqrt(6) A022840 A276856 A276872
sqrt(7) A022841 A276857 A276873
sqrt(8) A022842 A276858 A276874
e A022843 A276859 A276875
2*e A276853 A276860 A276876
Pi A022844 A063438 A276877
2*Pi A028130 A276861 A276878
1+sqrt(2) A003151 A276862 A276879
1+sqrt(3) A054088 A007538 A276880
1+sqrt(5) A276854 A276863 A276881
2+sqrt(2) A001952 A276864 A276882
2+sqrt(3) A003512 A276865 A276883
2+sqrt(5) A004976 A276866 A276884
1+tau A001950 2 + A003849 A276885
2+tau A003231 A276867 A276886
3+tau A276855 A276868 A276887
2+sqrt(1/2) A182769 A276869 A276888
sqrt(2)+sqrt(3) A110117 A276870 A276889
From Jeffrey Shallit, Aug 15 2023: (Start)
Simpler description: this sequence represents those positive integers that CANNOT be expressed as a difference of two elements of A022839.
There is a 20-state Fibonacci automaton for the terms of this sequence (see a276871.pdf). It takes as input the Zeckendorf representation of n and accepts iff n is a member of A276871. (End)

Examples

			The Beatty sequence for sqrt(5) is A022839 = (0,2,4,6,8,11,13,15,...), with difference sequence s = A081427 = (2,2,2,2,3,2,2,2,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,6,7,8,9,11,12,...), with complement (1,10,19,28,37,...).

Links

Crossrefs

Cf. A022839, A081427.

Programs

  • Mathematica
    z = 500; r = Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A022839 *)
t = Differences[b]; (* A081427 *)
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];  (* A276871 *)

A352676 Intersection of Beatty sequences for sqrt(3) and 1+sqrt(3).

Original entry on oeis.org

5, 8, 10, 13, 19, 24, 27, 32, 38, 43, 46, 51, 57, 60, 62, 65, 71, 76, 79, 81, 84, 90, 95, 98, 103, 109, 112, 114, 117, 122, 128, 131, 133, 136, 142, 147, 150, 152, 155, 161, 166, 169, 174, 180, 183, 185, 188, 193, 199, 202, 204, 207, 213, 218, 221, 226, 232
Offset: 1

Views

Author

Clark Kimberling, Mar 26 2022

Keywords

Comments

Conjectures:
(1) a(n+1)-a(n) is in {2,3,4,5,6} for every n, and each of these differences occurs infinitely many times.
(2) Limit_{n->oo} a(n)/n = (3/2)*(1+sqrt(3)).
(3) Let d(n) = a(n) - A352673(n); then d(n) = 0 for infinitely many n, but {d(n)} is unbounded below and above.

Examples

			The two Beatty sequences, (1,3,5,6,8,10,12,13,15,17,19,20,...) and (2,5,8,10,13,16,19,21,24,...), share the numbers (5,8,10,13,19,24,...).

Links

Crossrefs

Cf. A022838, A054088.

Programs

  • Mathematica
    z = 200; r = Sqrt[3]; s = 1 + Sqrt[3];
u = Table[Floor[n r], {n, 1, z}]    (* A022838 *)
v = Table[Floor[n s], {n, 1, z}]    (* A054088 *)
Intersection[u, v]  (* A352676 *)

A186540 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) before g(j) when f(i)=g(j), where f(i)=i^2 and g(j)=-2+3j^2. Complement of A186539.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 30, 32, 35, 38, 40, 43, 46, 49, 51, 54, 57, 60, 62, 65, 68, 71, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 112, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 142, 144, 147, 150, 152, 155, 158, 161, 163, 166, 169, 172, 174, 177, 180, 183, 185, 188, 191, 193, 196, 199, 202, 204, 207, 210, 213, 215, 218
Offset: 1

Views

Author

Clark Kimberling, Feb 23 2011

Keywords

Comments

See A186219 for a discussion of adjusted joint rank sequences.
Does this differ from A054088? The first 42000 entries of both sequences at least are the same. - R. J. Mathar, Feb 25 2011

Examples

			First, write
1..4..9..16..25..36..49.. (i^2)
.......10....25....46.. (-2+3j^2)
Then replace each number by its rank, where ties are settled by ranking i^2 before -2+3j^2:
b=(1,3,4,6,7,9,11,12,14,15,17,18,..)=A186539
a=(2,5,8,10,13,16,19,21,24,27,30...).

Crossrefs

Cf. A186219, A186539, A186541, A186542.

Programs

Formula

b(n) = n+floor(sqrt((1/3)n^2+1/24)) = A186539(n).
a(n) = n+floor(sqrt(3n^2-3/2)).

A054086 For k >= 1, let p(k)=least h in N not already an a(i), q(k)=p(k)+k, a(3k-1)=q(k), a(3k)=p(k), a(3k+1)=least h in N not already an a(i).

Original entry on oeis.org

1, 3, 2, 4, 7, 5, 6, 11, 8, 9, 14, 10, 12, 18, 13, 15, 22, 16, 17, 26, 19, 20, 29, 21, 23, 33, 24, 25, 37, 27, 28, 41, 30, 31, 44, 32, 34, 48, 35, 36, 52, 38, 39, 55, 40, 42, 59, 43, 45, 63, 46, 47, 67, 49, 50, 70, 51, 53, 74, 54, 56, 78, 57
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

The trisection (3, 7, 11, 14, ...) is A003512, with n-th term [n*(2+sqrt(3))], the complement of this is A003511, with n-th term [n*(1+sqrt(3))/2].
A054087, A054088.

A059556 Beatty sequence for 1 + 1/gamma.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 19, 21, 24, 27, 30, 32, 35, 38, 40, 43, 46, 49, 51, 54, 57, 60, 62, 65, 68, 71, 73, 76, 79, 81, 84, 87, 90, 92, 95, 98, 101, 103, 106, 109, 112, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 142, 144, 147, 150, 153, 155, 158, 161, 163, 166
Offset: 1

Views

Author

Mitch Harris, Jan 22 2001

Keywords

Comments

Differs from A054088 at indices 56, 71, 112, 127, 142, 168, 183 etc. - R. J. Mathar, Oct 05 2008
Let r = gamma (the Euler constant, 0.5772...). When {k*r, k >= 1} is jointly ranked with the positive integers, A059555(n) is the position of n and A059556(n) is the position of n*r. - Clark Kimberling, Oct 21 2014

Links

Crossrefs

Beatty complement is A059555.

Programs

  • Mathematica
    t = N[Table[k*EulerGamma, {k, 1, 200}]]; u = Union[Range[200], t]
Flatten[Table[Flatten[Position[u, n]], {n, 1, 100}]]  (* A059556 *)
Flatten[Table[Flatten[Position[u, t[[n]]]], {n, 1, 100}]] (* A059555 *)
(* Clark Kimberling, Oct 21 2014 *)
  • PARI
    { default(realprecision, 100); b=1 + 1/Euler; for (n = 1, 2000, write("b059556.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

A276880 Sums-complement of the Beatty sequence for 1 + sqrt(3).

Original entry on oeis.org

1, 4, 7, 12, 15, 18, 23, 26, 29, 34, 37, 42, 45, 48, 53, 56, 59, 64, 67, 70, 75, 78, 83, 86, 89, 94, 97, 100, 105, 108, 111, 116, 119, 124, 127, 130, 135, 138, 141, 146, 149, 154, 157, 160, 165, 168, 171, 176, 179, 182, 187, 190, 195, 198, 201, 206, 209, 212
Offset: 1

Views

Author

Clark Kimberling, Sep 27 2016

Keywords

Comments

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

Examples

			The Beatty sequence for 1 + sqrt(3) is A054088 = (0,2,5,8,19,13,16,...), with difference sequence s = A007538 = (2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,8,9,10,11,13,...), with complement (1,4,7,12,15,18,23,...).

Links

Crossrefs

Cf. A054088, A007538, A276871.

Programs

  • Mathematica
    z = 500; r = 1 + Sqrt[3]; b = Table[Floor[k*r], {k, 0, z}]; (* A054088 *)
t = Differences[b]; (* A007538 *)
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]  (* A276880 *)

A194140 a(n) = Sum_{j=1..n} floor(j*(1+sqrt(3))); n-th partial sum of Beatty sequence for 1+sqrt(3).

Original entry on oeis.org

2, 7, 15, 25, 38, 54, 73, 94, 118, 145, 175, 207, 242, 280, 320, 363, 409, 458, 509, 563, 620, 680, 742, 807, 875, 946, 1019, 1095, 1174, 1255, 1339, 1426, 1516, 1608, 1703, 1801, 1902, 2005, 2111, 2220, 2332, 2446, 2563, 2683, 2805, 2930, 3058
Offset: 1

Views

Author

Clark Kimberling, Aug 17 2011

Keywords

Crossrefs

Cf. A054088 (Beatty sequence for 1+sqrt(3)).

Programs

  • Mathematica
    c[n_] := Sum[Floor[j*(1+Sqrt[3])], {j, 1, n}];
c = Table[c[n], {n, 1, 90}]
  • Python
    from sympy import integer_nthroot
def A194140(n): return n*(n+1)//2+sum(integer_nthroot(3*j**2,2)[0] for j in range(1,n+1)) # Chai Wah Wu, Mar 17 2021

Extensions

Offset corrected by Alois P. Heinz, Mar 17 2021
Showing 1-7 of 7 results.

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