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A276871 Sums-complement of the Beatty sequence for sqrt(5). 19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

Table of n, a(n) for n=1..54.

Index entries for sequences related to Beatty sequences

EXAMPLE

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,...).

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 *)

CROSSREFS

Cf. A022839, A081427.

Sequence in context: A089756 A097153 A017173 * A247465 A088410 A179110

Adjacent sequences:  A276868 A276869 A276870 * A276872 A276873 A276874

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Sep 24 2016

STATUS

approved

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Last modified February 15 22:28 EST 2019. Contains 320138 sequences. (Running on oeis4.)