A194919 -id:A194919 - cp's OEIS frontend

A194917 Fractalization of (n-[nr-n]), where [ ]=floor and r=(1+sqrt(5))/2 (the golden ratio).

1, 2, 1, 2, 3, 1, 2, 4, 3, 1, 2, 5, 4, 3, 1, 2, 5, 6, 4, 3, 1, 2, 5, 7, 6, 4, 3, 1, 2, 5, 7, 8, 6, 4, 3, 1, 2, 5, 7, 9, 8, 6, 4, 3, 1, 2, 5, 7, 10, 9, 8, 6, 4, 3, 1, 2, 5, 7, 10, 11, 9, 8, 6, 4, 3, 1, 2, 5, 7, 10, 12, 11, 9, 8, 6, 4, 3, 1, 2, 5, 7, 10, 13, 12, 11, 9, 8, 6, 4, 3, 1, 2, 5, 7

Offset: 1

Clark Kimberling, Sep 08 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (n-[nr-n]) is A189663.

Cf. A194959, A189663, A194918, A194919.

r = GoldenRatio; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A189663 *)
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] (*  A194917 *)
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}]] (* A194918 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194919 *)

A194918 Interspersion fractally induced by A189663, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 7, 9, 8, 15, 11, 14, 13, 12, 21, 16, 20, 19, 17, 18, 28, 22, 27, 26, 23, 25, 24, 36, 29, 35, 34, 30, 33, 31, 32, 45, 37, 44, 43, 38, 42, 39, 41, 40, 55, 46, 54, 53, 47, 52, 48, 51, 50, 49, 66, 56, 65, 64, 57, 63, 58, 62, 61, 59, 60, 78, 67, 77

Offset: 1

Clark Kimberling, Sep 08 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194918 is a permutation of the positive integers, with inverse A194919.

			Northwest corner:
1...3...6...10..15..21
2...4...7...11..16..22
5...9...14..20..27..35
8...13..19..26..34..43
12..17..23..30..38..47

Cf. A194959, A189663, A194917, A194919.

r = GoldenRatio; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A189663 *)
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] (*  A194917 *)
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}]] (* A194918 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194919 *)

cp's OEIS Frontend

A194917 Fractalization of (n-[nr-n]), where [ ]=floor and r=(1+sqrt(5))/2 (the golden ratio).

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