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w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]; (* A276877 *)

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proposed

allocated Sums-complement of the Beatty sequence for Clark KimberlingPi.

1, 2, 5, 8, 11, 14, 17, 20, 23, 24, 27, 30, 33, 36, 39, 42, 45, 46, 49, 52, 55, 58, 61, 64, 67, 68, 71, 74, 77, 80, 83, 86, 89, 90, 93, 96, 99, 102, 105, 108, 111, 112, 115, 118, 121, 124, 127, 130, 133, 134, 137, 140, 143, 146, 149, 152, 155, 156, 159, 162

1,2

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

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

The Beatty sequence for Pi is A022844 = (0,3,6,9,12,15,18,21,25,,...), with difference sequence s = A063438 = (3,3,3,3,3,3,3,4,3,3,3,...). 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,...).

z = 500; r = Pi; b = Table[Floor[k*r], {k, 0, z}]; (* A022844 *)

t = Differences[b]; (* A063438 *)

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]; (* A276877 *)

Cf. A022844, A063438, A276871.

allocated

nonn,easy

Clark Kimberling, Sep 27 2016

approved

editing

allocated for Clark Kimberling

allocated

approved