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 1-4 of 4 results.

A194101 Inverse permutation of A194100; every positive integer occurs exactly once.

Original entry on oeis.org

1, 3, 6, 10, 15, 2, 5, 9, 14, 20, 21, 28, 4, 8, 13, 19, 26, 27, 35, 36, 45, 55, 7, 12, 18, 25, 33, 34, 43, 44, 54, 65, 66, 78, 91, 11, 17, 24, 32, 41, 42, 52, 53, 64, 76, 77, 90, 104, 105, 120, 16, 23, 31, 40, 50, 51, 62, 63, 75, 88, 89, 103, 118, 119, 135, 136, 22
Offset: 1

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Crossrefs

Cf. A194100.

Programs

A194104 Natural interspersion of A194102; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 12, 8, 9, 6, 19, 13, 14, 10, 11, 27, 20, 21, 15, 16, 17, 36, 28, 29, 22, 23, 24, 18, 47, 37, 38, 30, 31, 32, 25, 26, 59, 48, 49, 39, 40, 41, 33, 34, 35, 73, 60, 61, 50, 51, 52, 42, 43, 44, 45, 88, 74, 75, 62, 63, 64, 53, 54, 55, 56, 46, 104, 89, 90
Offset: 1

Views

Author

Clark Kimberling, Aug 15 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, A194100 is a permutation of the positive integers; its inverse is A194101.

Examples

			Northwest corner:
1...3...7...12...19
2...4...8...13...20
5...9...14..21...29
6...10..15..22...30
11..16..23..31...40

Crossrefs

Cf. A194029, A194102, A194103, A194105.

Programs

  • Mathematica
    z = 40; g = Sqrt[2];
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194102 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194103  new *)
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}]]  (* A194104 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194105 *)

A194126 -1+A088207.

Original entry on oeis.org

1, 6, 13, 23, 36, 51, 69, 89, 112, 138, 166, 197, 231, 267, 306, 347, 391, 438, 487, 539, 593, 650, 710, 772, 837, 905, 975, 1048, 1123, 1201, 1282, 1365, 1451, 1540, 1631, 1725, 1821, 1920, 2022, 2126, 2233, 2342, 2454, 2569, 2686, 2806, 2929
Offset: 1

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Comments

A194077 is the natural fractal sequence of A194126.

Crossrefs

Cf. A088207, A194077, A194100.

Programs

  • Mathematica
    c[k_]:=-1+Sum[Floor[j+j*GoldenRatio],{j,1,k}];
c=Table[c[k],{k,1,40}]

Formula

a(n)=-1+sum(floor(j+j*r) : 1<=j<=n), where r=(1+sqrt(5))/2, the golden ratio.

A193042 Natural fractal sequence of A194126.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 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, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
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, A194126, A194100.

Programs

  • Mathematica
    z = 40; g = GoldenRatio;
c[k_] := -1 + Sum[Floor[j + j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194126 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A193042 *)
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}]]  (* A194100 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194101 *)
Showing 1-4 of 4 results.

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