A110117 -id:A110117 - cp's OEIS frontend

A276870 First differences of the Beatty sequence A110117 for sqrt(2) + sqrt(3).

3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3

Offset: 1

Clark Kimberling, Sep 26 2016

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A110117, A276889.

[Floor(n*(Sqrt(2) + Sqrt(3))) - Floor((n-1)*(Sqrt(2) + Sqrt(3))): n in [1..100]]; // G. C. Greubel, Aug 16 2018

z = 500; r = Sqrt[2]+Sqrt[3]; b = Table[Floor[k*r], {k, 0, z}] (* A110117 *)
Differences[b] (* A276870 *)

vector(100, n, floor(n*(sqrt(2) + sqrt(3))) - floor((n-1)*(sqrt(2)+sqrt(3)))) \\ G. C. Greubel, Aug 16 2018

a(n) = floor(n*r) - floor(n*r - r), where r = sqrt(2) + sqrt(3), n >= 1.

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

Clark Kimberling, Sep 24 2016

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)

			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,...).

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.
Jeffrey Shallit, Fibonacci automaton for A276871
Index entries for sequences related to Beatty sequences

Cf. A022839, A081427.

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 *)

A172332 Floor(n*(sqrt(13)+sqrt(5))).

Original entry on oeis.org

0, 5, 11, 17, 23, 29, 35, 40, 46, 52, 58, 64, 70, 75, 81, 87, 93, 99, 105, 110, 116, 122, 128, 134, 140, 146, 151, 157, 163, 169, 175, 181, 186, 192, 198, 204, 210, 216, 221, 227, 233, 239, 245, 251, 257, 262, 268, 274, 280, 286, 292

Offset: 0

Vincenzo Librandi, Feb 01 2010

Also integer part of n*5.8416192529..., where the constant is the largest root of x^4 -36*x^2 +64.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A110117, A172323-A172332, A172334, A172336-A172338.

[Floor(n*(Sqrt(13)+Sqrt(5))): n in [0..60]];

With[{c = Sqrt[13] + Sqrt[5]}, Floor[c Range[0, 70]]] (* Vincenzo Librandi, Aug 01 2013 *)

A172334 Floor(n*(sqrt(13)+sqrt(3))).

Original entry on oeis.org

0, 5, 10, 16, 21, 26, 32, 37, 42, 48, 53, 58, 64, 69, 74, 80, 85, 90, 96, 101, 106, 112, 117, 122, 128, 133, 138, 144, 149, 154, 160, 165, 170, 176, 181, 186, 192, 197, 202, 208, 213, 218, 224, 229, 234, 240, 245, 250, 256, 261, 266

Offset: 0

Vincenzo Librandi, Feb 01 2010

Also integer part of n*5.3376020830..., where the constant is the largest root of x^4 -32*x^2 +100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A110117, A172323-A172332, A172336-A172338.

[ Floor(n*(Sqrt(13)+Sqrt(3))): n in [0..60] ];

With[{c = Sqrt[13] + Sqrt[3]}, Table[Floor[c n], {n, 0, 50}]] (* Harvey P. Dale, Apr 25 2011 *)

A172336 a(n) = floor(n*(sqrt(13)+sqrt(2))).

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 256, 261, 266, 271, 276

Offset: 0

Vincenzo Librandi, Feb 01 2010

a(n) = integer part of n*(sqrt(13)+sqrt(2)), where the constant is the largest root of x^4 -30*x^2 +121.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A110117, A172323-A172332, A172334, A172337, A172338.

[Floor(n*(Sqrt(13)+Sqrt(2))): n in [0..60]];

With[{c = Sqrt[13] + Sqrt[2]}, Floor[c Range[0, 70]]] (* Vincenzo Librandi, Aug 01 2013 *)

for(n=0,50, print1(floor(n*(sqrt(13)+sqrt(2))), ", ")) \\ G. C. Greubel, Jul 05 2017

A172338 a(n) = floor(n*(sqrt(5)+sqrt(3))).

Original entry on oeis.org

0, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218

Offset: 0

Vincenzo Librandi, Feb 01 2010

sqrt(5)+sqrt(3) = 3.96811878506867... is the largest root of x^4 - 16*x^2 + 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Beatty sequences

Cf. A110117, A172323-A172332, A172334, A172336, A172337.

[Floor(n*Sqrt(5)+Sqrt(3)): n in [0..60] ];

With[{c = Sqrt[5] + Sqrt[3]}, Floor[c Range[0,60]]] (* Harvey P. Dale, Apr 10 2012 *)

a(n)=sqrtint(sqrtint(60*n^4)+8*n^2) \\ Charles R Greathouse IV, Jan 24 2022

A110118 a(n) = floor(n*(sqrt(6) + sqrt(2) + 2)/4).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102

Offset: 1

Reinhard Zumkeller, Jul 13 2005

nonn

Beatty sequence for (sqrt(6)+sqrt(2)+2)/4 = 1.465925826289...; complement of A110117.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A110119.

[Floor(n*(Sqrt(6) + Sqrt(2) + 2)/4): n in [1..100]]; // G. C. Greubel, Mar 27 2018

With[{c = (Sqrt[2] + Sqrt[6] + 2)/4}, Floor[c*Range[100]]] (* G. C. Greubel, Mar 27 2018 *)

for(n=1,100, print1(floor(n*(sqrt(6) + sqrt(2) + 2)/4), ", ")) \\ G. C. Greubel, Mar 27 2018

A138281 a(n) = floor((sqrt(2) + sqrt(3))^n).

Original entry on oeis.org

1, 3, 9, 31, 97, 308, 969, 3051, 9601, 30210, 95049, 299052, 940897, 2960313, 9313929, 29304086, 92198401, 290080547, 912670089, 2871501385, 9034502497, 28424933309, 89432354889, 281377831710, 885289046401, 2785353383794, 8763458109129, 27572156006234

Offset: 0

Reinhard Zumkeller, Mar 12 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A089078, A110117, A121503, A135611, A135798.

[Floor((Sqrt(2) + Sqrt(3))^n): n in [0..50]]; // G. C. Greubel, Jan 27 2018

Table[Floor[(Sqrt[2] + Sqrt[3])^n], {n, 0, 50}] (* G. C. Greubel, Jan 27 2018 *)

for(n=0,50, print1(floor((sqrt(2) + sqrt(3))^n), ", ")) \\ G. C. Greubel, Jan 27 2018

a(2*n) = floor(A001079(n) + A001078(n)*sqrt(6));
(sqrt(2) + sqrt(3))^(2*n) = A001079(n) + A001078(n)*sqrt(6);
a(2*n+1) = floor(A054320(n)*sqrt(2) + A138288(n)*sqrt(3));
(sqrt(2)+sqrt(3))^(2*n+1) = A054320(n)*sqrt(2) + A138288(n)*sqrt(3).

Terms a(16) and a(18) corrected, terms a(19) onward added by G. C. Greubel, Jan 27 2018

A172337 Floor(n*(sqrt(11)+sqrt(7))).

Original entry on oeis.org

0, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292, 298, 304, 310, 316, 321, 327

Offset: 0

Vincenzo Librandi, Feb 01 2010

a(n) = integer part of n*(sqrt(11)+sqrt(7)), where the constant is the largest root of x^4 -36*x^2 +16.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A110117, A172323-A172332, A172334, A172336-A172338.

[ Floor(n*(Sqrt(11)+Sqrt(7))): n in [0..60] ];

With[{c = Sqrt[11] + Sqrt[7]}, Floor[c Range[0, 70]]] (* Vincenzo Librandi, Aug 01 2013 *)

A276889 Sums-complement of the Beatty sequence for sqrt(2) + sqrt(3).

Original entry on oeis.org

1, 2, 5, 8, 11, 14, 17, 20, 21, 24, 27, 30, 33, 36, 39, 42, 43, 46, 49, 52, 55, 58, 61, 64, 65, 68, 71, 74, 77, 80, 83, 86, 87, 90, 93, 96, 99, 102, 105, 108, 109, 112, 115, 118, 121, 124, 127, 130, 131, 134, 137, 140, 143, 146, 149, 150, 153, 156, 159, 162

Offset: 1

Clark Kimberling, Oct 01 2016

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

			The Beatty sequence for sqrt(2) + sqrt(3) is A110117 = (0,3,6,9,12,15,18,22,...), with difference sequence s = A276870 = (3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,...).  The sums s(j)+s(j+1)+...+s(k) include (3,4,6,7,9,10,...), with complement (1,2,5,8,11,14,17,20,21,...).

Index entries for sequences related to Beatty sequences

Cf. A110117, A276870, A276871.

z = 500; r = Sqrt[2] + Sqrt[3]; b = Table[Floor[k*r], {k, 0, z}]; (* A110117 *)
t = Differences[b]; (* A276870 *)
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];  (* A276889 *)

cp's OEIS Frontend

A276870 First differences of the Beatty sequence A110117 for sqrt(2) + sqrt(3).

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

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A172332 Floor(n*(sqrt(13)+sqrt(5))).

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A172334 Floor(n*(sqrt(13)+sqrt(3))).

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A172336 a(n) = floor(n*(sqrt(13)+sqrt(2))).

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A172338 a(n) = floor(n*(sqrt(5)+sqrt(3))).

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A110118 a(n) = floor(n*(sqrt(6) + sqrt(2) + 2)/4).

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A138281 a(n) = floor((sqrt(2) + sqrt(3))^n).

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