A194108 -id:A194108 - cp's OEIS frontend

A194109 Inverse permutation of A194108; every positive integer occurs exactly once.

1, 3, 6, 2, 5, 9, 10, 15, 4, 8, 13, 14, 20, 21, 7, 12, 18, 19, 26, 27, 28, 36, 11, 17, 24, 25, 33, 34, 35, 44, 45, 55, 16, 23, 31, 32, 41, 42, 43, 53, 54, 65, 66, 78, 22, 30, 39, 40, 50, 51, 52, 63, 64, 76, 77, 90, 91, 29, 38, 48, 49, 60, 61, 62, 74, 75, 88, 89, 103

Offset: 1

Clark Kimberling, Aug 15 2011

nonn

Cf. A194108.

Mathematica
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(See A194108.)
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A194106 Sum{floor(j*sqrt(3)) : 1<=j<=n}; n-th partial sum of Beatty sequence for sqrt(3).

Original entry on oeis.org

1, 4, 9, 15, 23, 33, 45, 58, 73, 90, 109, 129, 151, 175, 200, 227, 256, 287, 319, 353, 389, 427, 466, 507, 550, 595, 641, 689, 739, 790, 843, 898, 955, 1013, 1073, 1135, 1199, 1264, 1331, 1400, 1471, 1543, 1617, 1693, 1770, 1849, 1930, 2013, 2097

Offset: 1

Clark Kimberling, Aug 15 2011

nonn

Cf. A194107, A194108, A194109, A022838 (Beatty sequence for sqrt(3)).

c[n_] := Sum[Floor[j*Sqrt[3]], {j, 1, n}];
c = Table[c[n], {n, 1, 90}]

A194107 Natural fractal sequence of A194106.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

Offset: 1

Clark Kimberling, Aug 15 2011

nonn

See A194029 for definitions of natural fractal sequence and natural interspersion.

Cf. A194029, A194106, A194108.

z = 40; g = Sqrt[3];
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194106 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194107 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194108 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194109 *)

A195108 Interspersion fractally induced by A004736.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 8, 10, 7, 9, 13, 15, 11, 14, 12, 19, 21, 17, 20, 18, 16, 26, 28, 23, 27, 24, 22, 25, 34, 36, 30, 35, 32, 29, 33, 31, 43, 45, 39, 44, 41, 37, 42, 40, 38, 53, 55, 49, 54, 51, 47, 52, 50, 48, 46, 64, 66, 59, 65, 62, 57, 63, 61, 58, 56, 60, 76, 78, 71

Offset: 1

Clark Kimberling, Sep 09 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence.
The sequence A004736 is the fractal sequence obtained by concatenating the segments 1; 2,1; 3,2,1; 4,3,2,1;...
Every pair of rows of A195108 eventually intersperse.
As a sequence, A194108 is a permutation of the positive integers, with inverse A195109.

			Northwest corner:
1...2...5...8...13..19..26
3...6...10..15..21..28..36
4...7...11..17..23..30..39
9...14..20..27..35..44..54
12..18..24..32..41..51..62

Cf. A194959, A004736, A195107, A195109.

j[n_] := Table[n + 1 - k, {k, 1, n}]; t[1] = j[1];
t[n_] := Join[t[n - 1], j[n]]   (* A004736 *)
t[10]
p[n_] := t[20][[n]]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195107 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A195108 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A195109 *)

cp's OEIS Frontend

A194109 Inverse permutation of A194108; every positive integer occurs exactly once.

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A194106 Sum{floor(j*sqrt(3)) : 1<=j<=n}; n-th partial sum of Beatty sequence for sqrt(3).

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A194107 Natural fractal sequence of A194106.

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A195108 Interspersion fractally induced by A004736.

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