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 1-10 of 11 results. Next

A269974 Factorial-nested interval sequence of sqrt(1/2).

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 2, 1, 2, 3, 1, 3, 3, 1, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 1, 2
Offset: 1

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

See A269970 for a definition of factorial-nested interval sequence and guide to related sequences.

Crossrefs

Cf. A000142, A269970.

A269979 Decimal expansion of the number having (1,2,3,4,...) = A000027 as its factorial-nested interval sequence.

Original entry on oeis.org

5, 9, 0, 4, 5, 2, 3, 5, 4, 3, 5, 2, 0, 5, 5, 4, 8, 1, 6, 8, 1, 2, 4, 3, 2, 8, 1, 0, 1, 3, 5, 0, 2, 4, 2, 7, 9, 7, 1, 0, 4, 3, 5, 7, 7, 1, 7, 7, 7, 3, 5, 0, 0, 6, 3, 9, 6, 4, 8, 3, 9, 1, 0, 6, 9, 9, 8, 9, 2, 0, 1, 5, 6, 0, 3, 8, 8, 8, 9, 8, 9, 4, 6, 7, 7, 0
Offset: 0

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

Suppose that r = (r(n)) is a sequence satisfying (i) 1 = r(1) > r(2) > r(3) > ... and (ii) r(n) -> 0. For x in (0,1], let n(1) be the index n such that r(n+1) , x <= r(n), and let L(1) = r(n(1))-r(n(1)+1). Let n(2) be the index n such that r(n(1)+1) < x <= r(n(1)+1) + L(1)r(n), and let L(2) = (r(n(2))-r(r(n)+1)L(1). Continue inductively to obtain the sequence (n(1), n(2), n(3), ... ), the r-nested interval sequence of x. Taking r = (1/n!) gives the factorial-nested interval sequence of x.
Conversely, given a sequence s= (n(1),n(2),n(3),...) of positive integers, the number x having satisfying NI(x) = s is the sum of left-endpoints of nested intervals (r(n(k)+1), r(n(k))]; i.e., x = sum{L(k)r(n(k+1)+1), k >=1}, where L(0) = 1.
See A269970 for a guide to related sequences.

Examples

			x = 0.59045235435205548168124328101350...

Crossrefs

Cf. A000027, A000142, A269970.

Programs

  • Mathematica
    r[n_] := 1/n!; n[k_] := k; Table[n[k], {k, 1, 1000}];
len[1] = r[n[1]] - r[n[1] + 1];
len[k_] := len[k - 1]*(r[n[k]] - r[n[k] + 1])
sum = r[n[1] + 1] + Sum[len[i]*r[n[i + 1] + 1], {i, 1, 300}];
g = N[sum, 150]
RealDigits[g, 10, 100][[1]]

A269971 Factorial-nested interval sequence of e-2.

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 1, 1, 2, 5, 2, 1, 2, 3, 1, 3, 2, 1, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 2, 2, 2, 1, 1, 3, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

See A269970 for a definition of factorial-nested interval sequence and guide to related sequences.

Crossrefs

Cf. A000142, A269970.

A269972 Factorial-nested interval sequence of 1/Pi.

Original entry on oeis.org

2, 2, 1, 1, 2, 1, 1, 1, 4, 2, 1, 3, 3, 2, 1, 1, 3, 2, 1, 2, 3, 2, 1, 1, 3, 1, 3, 1, 1, 2, 1, 2, 1, 3, 1, 1, 2, 1, 2, 3, 2, 3, 4, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 2, 1, 5, 3, 2, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 4, 3, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

See A269970 for a definition of factorial-nested interval sequence and guide to related sequences.

Crossrefs

Cf. A000142, A269970.

A269973 Factorial-nested interval sequence of Pi-3.

Original entry on oeis.org

3, 1, 1, 2, 3, 2, 1, 2, 1, 1, 1, 4, 2, 3, 2, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 4, 1, 1, 1, 3, 3, 2, 1, 1, 4, 2, 1, 3, 2, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 2, 6, 2, 1, 2, 3, 1, 1, 2, 2, 2, 3, 3, 1, 1, 1, 2, 2, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

See A269970 for a definition of factorial-nested interval sequence and guide to related sequences.

Crossrefs

Cf. A000142, A269970.

A269975 Factorial-nested interval sequence of -1 + sqrt(2).

Original entry on oeis.org

2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 2, 1, 2, 3, 1, 3, 3, 1, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 1, 2, 1
Offset: 1

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

See A269970 for a definition of factorial-nested interval sequence and guide to related sequences.

Crossrefs

Cf. A000142, A269970.

Formula

Empirical: a(n) = A269974(n+1). - R. J. Mathar, Mar 21 2016

A269976 Factorial-nested interval sequence of sqrt(1/3).

Original entry on oeis.org

1, 3, 1, 1, 1, 2, 2, 3, 1, 3, 3, 2, 1, 1, 3, 2, 1, 4, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 4, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 1, 1, 5, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

See A269970 for a definition of factorial-nested interval sequence and guide to related sequences.

Crossrefs

Cf. A000142, A269970.

A269977 Factorial-nested interval sequence of -1 + sqrt(3).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 2, 2, 1, 3, 1, 3, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 3, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 3, 3, 1, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

See A269970 for a definition of factorial-nested interval sequence and guide to related sequences.

Crossrefs

Cf. A000142, A269970.

A269978 Factorial-nested interval sequence of 1/tau, where tau = golden ratio (A001622).

Original entry on oeis.org

1, 2, 2, 3, 1, 2, 2, 1, 1, 1, 1, 3, 1, 4, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 3, 2, 3, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 1, 2, 2, 4, 3, 3, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 3, 2, 3, 1, 2, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

See A269970 for a definition of factorial-nested interval sequence and guide to related sequences.

Crossrefs

Cf. A000142, A269970.

A269980 Decimal expansion of the number having (1,3,5,7,9,...) = A005408 as its factorial-nested interval sequence.

Original entry on oeis.org

5, 2, 0, 9, 2, 0, 1, 4, 9, 6, 5, 3, 4, 8, 7, 4, 7, 5, 7, 6, 2, 2, 8, 1, 9, 8, 9, 1, 1, 8, 7, 4, 3, 3, 7, 5, 4, 8, 1, 4, 5, 7, 9, 0, 7, 6, 5, 4, 9, 6, 8, 3, 6, 7, 1, 8, 3, 5, 7, 1, 7, 3, 6, 0, 5, 6, 5, 6, 3, 6, 0, 0, 1, 4, 3, 3, 2, 3, 4, 6, 3, 9, 4, 6, 6, 0
Offset: 0

Views

Author

Clark Kimberling, Mar 08 2016

Keywords

Comments

Suppose that r = (r(n)) is a sequence satisfying (i) 1 = r(1) > r(2) > r(3) > ... and (ii) r(n) -> 0. For x in (0,1], let n(1) be the index n such that r(n+1) , x <= r(n), and let L(1) = r(n(1))-r(n(1)+1). Let n(2) be the index n such that r(n(1)+1) < x <= r(n(1)+1) + L(1)r(n), and let L(2) = (r(n(2))-r(r(n)+1)L(1). Continue inductively to obtain the sequence (n(1), n(2), n(3), ... ), the r-nested interval sequence of x. Taking r = (1/n!) gives the factorial-nested interval sequence of x.
Conversely, given a sequence s= (n(1),n(2),n(3),...) of positive integers, the number x having satisfying NI(x) = s is the sum of left-endpoints of nested intervals (r(n(k)+1), r(n(k))]; i.e., x = sum{L(k)r(n(k+1)+1), k >=1}, where L(0) = 1.
See A269970 for a guide to related sequences.

Examples

			x = 0.5209201496534874757622819891187433754...

Crossrefs

Cf. A000142, A005408, A269970, A269979.

Programs

  • Mathematica
    r[n_] := 1/n!; n[k_] := 2 k -1; Table[n[k], {k, 1, 1000}];
len[1] = r[n[1]] - r[n[1] + 1];
len[k_] := len[k - 1]*(r[n[k]] - r[n[k] + 1])
sum = r[n[1] + 1] + Sum[len[i]*r[n[i + 1] + 1], {i, 1, 300}];
g = N[sum, 150]
RealDigits[g, 10, 100][[1]]
Showing 1-10 of 11 results. Next

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