A276879
Sums-complement of the Beatty sequence for 1 + sqrt(2).
Original entry on oeis.org
1, 6, 11, 18, 23, 30, 35, 40, 47, 52, 59, 64, 69, 76, 81, 88, 93, 100, 105, 110, 117, 122, 129, 134, 139, 146, 151, 158, 163, 170, 175, 180, 187, 192, 199, 204, 209, 216, 221, 228, 233, 238, 245, 250, 257, 262, 269, 274, 279, 286, 291, 298, 303, 308, 315
Offset: 1
The Beatty sequence for 1 + sqrt(2) is A003151 = (0,2,4,7,9,12,14,16,...), with difference sequence s = A276862 = (2,2,3,2,3,2,2,3,2,3,2,2,3,2,3,...). The sums s(j)+s(j+1)+...+s(k) include (2,3,4,5,7,8,9,12,...), with complement (1,6,11,18,23,...).
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z = 500; r = 1+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A003151 *)
t = Differences[b]; (* A276862 *)
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] (* A276879 *)
A276882
Sums-complement of the Beatty sequence for 2 + sqrt(2).
Original entry on oeis.org
1, 2, 5, 8, 9, 12, 15, 16, 19, 22, 25, 26, 29, 32, 33, 36, 39, 42, 43, 46, 49, 50, 53, 56, 57, 60, 63, 66, 67, 70, 73, 74, 77, 80, 83, 84, 87, 90, 91, 94, 97, 98, 101, 104, 107, 108, 111, 114, 115, 118, 121, 124, 125, 128, 131, 132, 135, 138, 141, 142, 145
Offset: 1
The Beatty sequence for 2 + sqrt(2) is A001952 = (0,3,6,10,13,17,20, 23,27,...), with difference sequence s = A276864 = (3,3,4,3,4,3,3,4,3,4,3,3,4,3,4,...). The sums s(j)+s(j+1)+...+s(k) include (3,4,6,7,10,11,13,14,17,...), with complement (1,2,5,8,9,12,15,...).
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z = 500; r = 2+Sqrt[2]; b = Table[Floor[k*r], {k, 0, z}]; (* A001952 *)
t = Differences[b]; (* A276864 *)
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] (* A276882 *)
A276883
Sums-complement of the Beatty sequence for 2 + sqrt(3).
Original entry on oeis.org
1, 2, 5, 6, 9, 10, 13, 16, 17, 20, 21, 24, 25, 28, 31, 32, 35, 36, 39, 40, 43, 46, 47, 50, 51, 54, 57, 58, 61, 62, 65, 66, 69, 72, 73, 76, 77, 80, 81, 84, 87, 88, 91, 92, 95, 96, 99, 102, 103, 106, 107, 110, 113, 114, 117, 118, 121, 122, 125, 128, 129, 132
Offset: 1
The Beatty sequence for 2 + sqrt(3) is A003512 = (0,3,7,11,14,18,22,26,...), with difference sequence s = A276865 = (3,4,4,3,4,4,4,3,4,4,4,3,4,4,3,...). The sums s(j)+s(j+1)+...+s(k) include (3,4,7,8,11,12,14,15,18,...), with complement (1,2,5,6,9,10,13,...).
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z = 500; r = 2 + Sqrt[3]; b = Table[Floor[k*r], {k, 0, z}]; (* A003512 *)
t = Differences[b]; (* A276865 *)
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] (* A276883 *)
A276884
Sums-complement of the Beatty sequence for 2 + sqrt(5).
Original entry on oeis.org
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
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
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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 *)
A276887
Sums-complement of the Beatty sequence for 3 + tau.
Original entry on oeis.org
1, 2, 3, 6, 7, 8, 11, 12, 15, 16, 17, 20, 21, 22, 25, 26, 29, 30, 31, 34, 35, 38, 39, 40, 43, 44, 45, 48, 49, 52, 53, 54, 57, 58, 59, 62, 63, 66, 67, 68, 71, 72, 75, 76, 77, 80, 81, 82, 85, 86, 89, 90, 91, 94, 95, 98, 99, 100, 103, 104, 105, 108, 109, 112
Offset: 1
The Beatty sequence for 3 + tau is A276855 = (-,4,9,13,18,23,27,...), with difference sequence s = A276868 = (4,5,4,5,5,4,5,4,5,5,4,5,5,4,5,4,...). The sums s(j)+s(j+1)+...+s(k) include (4,5,9,10,13,14,18,...), with complement (1,2,3,6,7,8,11,12,15,...).
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z = 500; r = 3 + GoldenRatio; b = Table[Floor[k*r], {k, 0, z}]; (* A276855 *)
t = Differences[b]; (* A276868 *)
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]; (* A276887 *)
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
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,...).
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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 *)
A276877
Sums-complement of the Beatty sequence for Pi.
Original entry on oeis.org
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
Offset: 1
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,...).
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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 *)
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
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,...).
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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 *)
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