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.

Previous Showing 31-39 of 39 results.

A194031 Inverse permutation of A194030; contains every positive integer exactly once.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 6, 11, 8, 9, 10, 15, 16, 12, 13, 14, 20, 21, 28, 36, 22, 17, 18, 19, 26, 27, 35, 44, 45, 55, 66, 78, 91, 29, 23, 24, 25, 33, 34, 43, 53, 54, 65, 77, 90, 37, 30, 31, 32, 41, 42, 52, 63, 64, 76
Offset: 1

Views

Author

Clark Kimberling, Aug 12 2011

Keywords

Comments

See A194029.

Links

Crossrefs

Cf. A194029, A194030 (inverse).

Programs

A194046 Natural interspersion of A052905, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 5, 2, 10, 6, 3, 16, 11, 7, 4, 23, 17, 12, 8, 9, 31, 24, 18, 13, 14, 15, 40, 32, 25, 19, 20, 21, 22, 50, 41, 33, 26, 27, 28, 29, 30, 61, 51, 42, 34, 35, 36, 37, 38, 39, 73, 62, 52, 43, 44, 45, 46, 47, 48, 49, 86, 74, 63, 53, 54, 55, 56, 57, 58, 59, 60, 100, 87, 75
Offset: 1

Views

Author

Clark Kimberling, Aug 13 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194046 is a permutation of the positive integers; its inverse is A194047.

Examples

			 Northwest corner:
1...5...10...16...23
2...6...11...17...24
3...7...12...18...25
4...8...13...19...26
9...14..20...27...35

Crossrefs

Cf. A194029, A194047

Programs

  • Mathematica
    z = 30;
c[k_] := (k^2 + 5 k - 4)/2;
c = Table[c[k], {k, 1, z}]  (* A052905 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* fractal sequence [A002260] *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194046 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194047 *)

A194048 Natural interspersion of A000330, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 5, 2, 14, 6, 3, 30, 15, 7, 4, 55, 31, 16, 8, 9, 91, 56, 32, 17, 18, 10, 140, 92, 57, 33, 34, 19, 11, 204, 141, 93, 58, 59, 35, 20, 12, 285, 205, 142, 94, 95, 60, 36, 21, 13, 385, 286, 206, 143, 144, 96, 61, 37, 22, 23
Offset: 1

Views

Author

Clark Kimberling, Aug 13 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194048 is a permutation of the positive integers; its inverse is A194049.

Examples

			Northwest corner:
1...5...14...30...55
2...6...15...31...56
3...7...16...32...57
4...8...17...33...58
9...18..34...59...95

Crossrefs

Cf. A194029, A194049.

Programs

  • Mathematica
    Remove["Global`*"];
z = 30;
c[k_] := k (k + 1) (2 k + 1)/6;
c = Table[c[k], {k, 1, z}]  (* A000330 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 500}]  (* fractal sequence [A064866] *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 10}, {k, 1, n}]]  (* A194048 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194049 *)

A194050 Natural fractal sequence of A014739.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Offset: 1

Views

Author

Clark Kimberling, Aug 13 2011

Keywords

Comments

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

Crossrefs

Cf. A194029, A194051, A194052.

Programs

  • Mathematica
    z = 50;
c[k_] := LucasL[k + 1] - 2;
c = Table[c[k], {k, 1, z}]  (* A014739 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 600}]  (* A194050 *)
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, 12}, {k, 1, n}]] (* A194051 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]] (* A194052 *)

A194053 Natural fractal sequence of A054347.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Comments

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

Crossrefs

Cf. A194029.

Programs

  • Mathematica
    z = 40; g = GoldenRatio
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A054347 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194053 *)
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}]]  (* A194054 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194058 *)

A194059 Natural interspersion of A001911 (Fibonacci numbers minus 2); a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 11, 7, 8, 9, 19, 12, 13, 14, 10, 32, 20, 21, 22, 15, 16, 53, 33, 34, 35, 23, 24, 17, 87, 54, 55, 56, 36, 37, 25, 18, 142, 88, 89, 90, 57, 58, 38, 26, 27, 231, 143, 144, 145, 91, 92, 59, 39, 40, 28, 375, 232, 233, 234, 146, 147, 93, 60, 61, 41, 29
Offset: 1

Views

Author

Clark Kimberling, Aug 14 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194059 is a permutation of the positive integers; its inverse is A194060.

Examples

			Northwest corner:
1...3...6...11...19
2...4...7...12...30
5...8...13..21...34
9...14..22..35...56
10..15..23..36...57

Crossrefs

Cf. A194029, A194059, A194062.

Programs

  • Mathematica
    z = 50;
c[k_] := -2 + Fibonacci[k + 3];
c = Table[c[k], {k, 1, z}]  (* A001911, F(n+3)-2 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 700}]   (* cf. A194055 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A194059 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 100}]] (* A194060 *)

A194063 Natural fractal sequence of A006578.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 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
Offset: 1

Views

Author

Clark Kimberling, Aug 14 2011

Keywords

Comments

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

Crossrefs

Cf. A194029, A006578, A194064, A194065.

Programs

  • Mathematica
    z = 50;
c[k_] := k (k + 1)/2 + Floor[(k^2)/4];
c = Table[c[k], {k, 1, z}]  (* A006578 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 400}]   (* A194063 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 11}, {k, 1, n}]] (* A194064 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]]  (* A194065 *)

A194066 Natural fractal sequence of A087483.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 14 2011

Keywords

Comments

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

Crossrefs

Cf. A194029, A194067, A087483.

Programs

  • Mathematica
    z = 70;
c[k_] := 1 + Floor[(1/3) k^2];
c = Table[c[k], {k, 1, z}]  (* A087483 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194066 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194067 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]] (* A194068 *)

A333907 For n >= 1, a(n) = Sum_{k=1..n} prevfib(k) + nextfib(k) - 2*k, where prevfib(k) is the largest Fibonacci number < k, nextfib(k) is the smallest Fibonacci number > k.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 2, 4, 7, 8, 7, 4, 7, 13, 17, 19, 19, 17, 13, 7, 12, 23, 32, 39, 44, 47, 48, 47, 44, 39, 32, 23, 12, 20, 39, 56, 71, 84, 95, 104, 111, 116, 119, 120, 119, 116, 111, 104, 95, 84, 71, 56, 39, 20, 33, 65, 95, 123, 149, 173, 195, 215, 233, 249, 263, 275
Offset: 1

Views

Author

Ctibor O. Zizka, Apr 09 2020

Keywords

Examples

			a(1) = (0 + 2 - 2*1) = 0;
a(2) = (0 + 2 - 2*1) + (1 + 3 - 2*2) = 0;
a(3) = (0 + 2 - 2*1) + (1 + 3 - 2*2) + (2 + 5 - 2*3) = 1;
a(4) = (0 + 2 - 2*1) + (1 + 3 - 2*2) + (2 + 5 - 2*3) + (3 + 5 - 2*4) = 1.

Crossrefs

Cf. A000045, A001076, A087172, A130473, A194029, A256654, A280514.

Programs

  • PARI
    isfib(k) = my(m=5*k^2); issquare(m-4) || issquare(m+4);
nextfib(n) = my(k=n+1); while (!isfib(k), k++); k;
prevfib(n) = my(k=n-1); while (!isfib(k), k--); k;
a(n) = sum(k=1, n, prevfib(k) + nextfib(k) - 2*k); \\ Michel Marcus, Apr 10 2020
Previous Showing 31-39 of 39 results.

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