The On-Line Encyclopedia of Integer Sequences (OEIS)

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A276884

Sums-complement of the Beatty sequence for 2 + sqrt(5).
(history; published version)

#10 by Bruno Berselli at Mon Jan 30 02:49:49 EST 2017

STATUS

reviewed

approved

#9 by Joerg Arndt at Mon Jan 30 02:45:16 EST 2017

STATUS

proposed

reviewed

#8 by Joerg Arndt at Mon Jan 30 02:45:12 EST 2017

STATUS

editing

proposed

#7 by Joerg Arndt at Mon Jan 30 02:45:01 EST 2017

EXAMPLE

The Beatty sequence for 2 + sqrt(5) is A004976 = (0,4,8,12,16,21,25,29, 33,38,42,46,50,55,59,63,...) with difference sequence s = A276866 = (4,4,4,4,5,4,4,4,5,4,4,4,5,4,4,...). The sums s(j)+s(j+1)+...+s(k) include (4,5,8,9,12,13,16,...), with complement (1,2,3,6,7,10,11,14,...). - corrected by _Michel Dekking_, Jan 30 2017

EXTENSIONS

Corrected by Michel Dekking, Jan 30 2017

STATUS

proposed

editing

#6 by Michel Dekking at Mon Jan 30 01:54:39 EST 2017

STATUS

editing

proposed

#5 by Michel Dekking at Mon Jan 30 01:53:11 EST 2017

EXAMPLE

The Beatty sequence for 2 + sqrt(5) is A004976 = (0,3,7,11,14,18,22,26,4,8,12,16,21,25,29, 33,38,42,46,50,55,59,63,...), with difference sequence s = A276866 = (3,4,4,3,4,4,4,3,5,4,4,4,3,5,4,4,3,4,5,4,4,...). The sums s(j)+s(j+1)+...+s(k) include (4,5,8,9,12,13,16,...), with complement (1,2,3,6,7,10,11,14,...).

EXTENSIONS

Corrected by Michel Dekking, Jan 30 2017

STATUS

approved

editing

Discussion

Mon Jan 30

01:54

Michel Dekking: Another sequence that had to be corrected.....

#4 by N. J. A. Sloane at Sun Oct 02 00:36:28 EDT 2016

STATUS

proposed

approved

#3 by Clark Kimberling at Sat Oct 01 17:23:50 EDT 2016

STATUS

editing

proposed

#2 by Clark Kimberling at Sat Oct 01 12:13:48 EDT 2016

NAME

allocated Sums-complement of the Beatty sequence for Clark Kimberling2 + sqrt(5).

DATA

1, 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 20, 23, 24, 27, 28, 31, 32, 35, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 54, 57, 58, 61, 62, 65, 66, 69, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 92, 95, 96, 99, 100, 103, 104, 107, 108, 109, 112, 113, 116, 117

OFFSET

1,2

COMMENTS

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

LINKS

<a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

EXAMPLE

The Beatty sequence for 2 + sqrt(5) is A004976 = (0,3,7,11,14,18,22,26,29,...), with difference sequence s = A276866 = (3,4,4,3,4,4,4,3,4,4,4,3,4,4,3,4,4,4,...). The sums s(j)+s(j+1)+...+s(k) include (4,5,8,9,12,13,16,...), with complement (1,2,3,6,7,10,11,14,...).

MATHEMATICA

z = 500; r = 2 + Sqrt[5]; b = Table[Floor[k*r], {k, 0, z}]; (* A004076 *)

t = Differences[b]; (* A276866 *)

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]; (* A276884 *)

CROSSREFS

Cf. A004976, A276866, A276871.

KEYWORD

allocated

nonn,easy

AUTHOR

Clark Kimberling, Oct 01 2016

STATUS

approved

editing

#1 by Clark Kimberling at Wed Sep 21 16:11:40 EDT 2016

NAME

allocated for Clark Kimberling

KEYWORD

allocated

STATUS

approved