cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A194922 Interspersion fractally induced by A194920, a rectangular array, by antidiagonals.

Original entry on oeis.org

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
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

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.

Examples

			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

Links

Crossrefs

Cf. A194920, A194921, A195071.

Programs

  • 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 *)

A195074 Interspersion fractally induced by A194920, a rectangular array, by antidiagonals.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

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, A194974 is a permutation of the positive integers, with inverse A195075.
To see that A195074 differs from A194988, note that the generating sequences A195072 and A194986 differ.

Examples

			Northwest corner:
1...3...6...10..15..21
2...4...7...11..16..22
5...9...14..20..27..35
8...12..17..23..30..38
13..19..26..34..43..53

Crossrefs

Cf. A195072, A195074, A195075.

Programs

  • Mathematica
    r = Sqrt[3]; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]   (* A195072 *)
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]   (* A195073 *)
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}]]    (* A195074 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]   (* A195075 *)

A194921 Fractalization of (n - [n/sqrt(2)]), where [ ]=floor.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (n-[n/sqrt(2)]) is A194920.

Links

Crossrefs

Cf. A194920, A194922, A195071.

Programs

  • 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 *)

A195071 Inverse permutation of A194922; every positive integer occurs exactly once.

Original entry on oeis.org

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
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Links

Crossrefs

Cf. A194922, A194921, A194920, A194959.

Programs

  • 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 *)

A255195 Triangle describing the shape of one eighth of the Gauss circle problem.

Original entry on oeis.org

1, 2, 0, 2, 1, 0, 2, 1, 1, 0, 2, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 1, 2, 2, 0, 0, 0, 2, 1, 1, 2, 1, 2, 0, 0, 0, 2, 1, 1, 1, 2, 2, 1, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0, 2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Mats Granvik, Feb 16 2015

Keywords

Comments

The sum of terms of row n is n.
Total of partial sums in reverse (from right to left) equals one eighth of the Gauss circle problem. Whenever there is the number 2 the border of the circle makes a jump upwards. Predicting where the 2's are would say something about the Gauss circle problem. The number of 2's equals the number of 0's in the same row, and is counted by A194920(n-1).

Examples

			1,
2, 0,
2, 1, 0,
2, 1, 1, 0,
2, 1, 2, 0, 0,
2, 1, 1, 2, 0, 0,
2, 1, 1, 2, 1, 0, 0,
2, 1, 1, 2, 2, 0, 0, 0,
2, 1, 1, 2, 1, 2, 0, 0, 0,
2, 1, 1, 1, 2, 2, 1, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 1, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 2, 0, 0, 0, 0,
2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 0, 0, 0

Links

Crossrefs

Cf. A000603, A194920.

Programs

  • Mathematica
    Flatten[Table[Sum[Table[If[And[If[n^2 + k^2 <= r^2, If[n >= k, 1, 0], 0] == 1, If[(n + 1)^2 + (k + 1)^2 <= r^2, If[n >= k, 1, 0], 0]== 0], 1, 0], {k, 0, r}], {n, 0, r}], {r, 0, 12}]]

Formula

A000603(n) = 2*(Sum_{k=1..n} Sum_{k=1..k} T(n,n-k+1))-ceiling((n-1)/sqrt(2)) for n>1.
A247588(n-1) = (Sum_{k=1..n} Sum_{k=1..k} (T(n,k) - T(n,n-k+1))/2).
Showing 1-5 of 5 results.

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