A194921 -id:A194921

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A194920

a(n) = n - floor(n/sqrt(2)).

+10
6

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22, 23

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

a(n) is the number of zeros in row n+1 in triangle A255195. - Mats Granvik, Feb 18 2015.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

MATHEMATICA

r = Sqrt[2]; p[n_] := n - Floor[n/r]

Table[p[n], {n, 1, 90}] (* A194920 *)

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] (* A194921 *)

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}]] (* A194922 *)

q[n_] := Position[w, n]; Flatten[Table[q[n],

{n, 1, 80}]] (* A195071 *)

PROG

(Magma) [n-Floor(n/Sqrt(2)): n in [1..80] ]; // Vincenzo Librandi, Sep 10 2011

(PARI) vector(100, n, n-floor(n/sqrt(2))) \\ Derek Orr, Feb 28 2015

CROSSREFS

Cf. A194921.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Sep 08 2011

STATUS

approved

A194922

Interspersion fractally induced by A194920, a rectangular array, by antidiagonals.

+10
4

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194922 is a permutation of the positive integers, with inverse A195071.

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

EXAMPLE

Northwest corner:

1, 3, 6, 10, 15, 21

2, 5, 9, 14, 20, 27, 35

4, 7, 11, 16, 22, 29, 37

8, 13, 19, 26, 34, 43, 53

12, 18, 25, 33, 42, 52, 63

MATHEMATICA

r = Sqrt[2]; p[n_] := n - Floor[n/r]

Table[p[n], {n, 1, 90}] (* A194920 *)

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] (* A194921 *)

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}]] (* A194922 *)

q[n_] := Position[w, n]; Flatten[Table[q[n],

{n, 1, 80}]] (* A195071 *)

CROSSREFS

Cf. A194920, A194921, A195071.

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Sep 08 2011

STATUS

approved

A195071

Inverse permutation of A194922; every positive integer occurs exactly once.

+10
4

1, 3, 2, 6, 5, 4, 9, 10, 8, 7, 13, 15, 14, 12, 11, 18, 21, 20, 19, 17, 16, 24, 27, 28, 26, 25, 23, 22, 31, 34, 36, 35, 33, 32, 30, 29, 39, 42, 45, 44, 43, 41, 40, 38, 37, 48, 51, 55, 54, 53, 52, 50, 49, 47, 46, 58, 61, 65, 66, 64, 63, 62, 60, 59, 57, 56, 69, 72, 76

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

MATHEMATICA

r = Sqrt[2]; p[n_] := n - Floor[n/r]

Table[p[n], {n, 1, 90}] (* A194920 *)

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] (* A194921 *)

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}]] (* A194922 *)

q[n_] := Position[w, n]; Flatten[Table[q[n],

{n, 1, 80}]] (* A195071 *)

CROSSREFS

Cf. A194922, A194921, A194920, A194959.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Sep 08 2011

STATUS

approved

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