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.

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A330117 Beatty sequence for 1+x, where 1/(1+x) + 1/(1+x+x^2) = 1.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 21, 22, 24, 26, 28, 29, 31, 33, 35, 36, 38, 40, 42, 43, 45, 47, 49, 50, 52, 54, 56, 57, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 107, 108, 110, 112
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2020

Keywords

Comments

Let x be the positive solution of 1/(1+x) + 1/(1+x+x^2) = 1. Then (floor(n*(1+x))) and (floor(n*(1+x+x^2))) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A075778, A329825, A330118 (complement).

Programs

  • Mathematica
    r = x /. FindRoot[1/(1 + x) + 1/(1 + x + x^2) == 1, {x, 1, 2}, WorkingPrecision -> 200]
RealDigits[r] (* A075778 *)
Table[Floor[n*(1 + r)], {n, 1, 250}]  (* A330117 *)
Table[Floor[n*(1 + r + r^2)], {n, 1, 250}]  (* A330118 *)
Plot[1/(1 + x) + 1/(1 + x + x^2) - 1, {x, 0, 2}]

Formula

a(n) = floor(n*(1+x)), where x = 0.7548776662... is the constant in A075778.

A330118 Beatty sequence for 1+x+x^2, where 1/(1+x) + 1/(1+x+x^2) = 1.

Original entry on oeis.org

2, 4, 6, 9, 11, 13, 16, 18, 20, 23, 25, 27, 30, 32, 34, 37, 39, 41, 44, 46, 48, 51, 53, 55, 58, 60, 62, 65, 67, 69, 72, 74, 76, 79, 81, 83, 86, 88, 90, 92, 95, 97, 99, 102, 104, 106, 109, 111, 113, 116, 118, 120, 123, 125, 127, 130, 132, 134, 137, 139, 141
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2020

Keywords

Comments

Let x be the positive solution of 1/(1+x) + 1/(1+x+x^2) = 1. Then (floor(n*(1+x))) and (floor(n*(1+x+x^2))) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A075778, A330117 (complement).

Programs

  • Mathematica
    r = x /. FindRoot[1/(1 + x) + 1/(1 + x + x^2) == 1, {x, 1, 2}, WorkingPrecision -> 200]
RealDigits[r] (* A075778 *)
Table[Floor[n*(1 + r)], {n, 1, 250}]  (* A330117 *)
Table[Floor[n*(1 + r + r^2)], {n, 1, 250}]  (* A330118 *)
Plot[1/(1 + x) + 1/(1 + x + x^2) - 1, {x, 0, 2}]

Formula

a(n) = floor(n*(1+x+x^2)), where x = 0.7548776662... is the constant in A075778.

A218197 Decimal expansion of the Perrin argument a (see below).

Original entry on oeis.org

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

Views

Author

Roman Witula, Oct 23 2012

Keywords

Comments

The Perrin argument a is defined by the decomposition of the known Perrin polynomial: X^3 - X - 1 = (X - t^(-1))*(X - i*sqrt(t)*e^(i*a))*(X + i*sqrt(t)*e^(-i*a)), where t = 0.754877666... (see A075778 and A060006 for the decimal expansions of t and t^(-1) respectively) is the only positive root of the polynomial x^3 + x^2 - 1 and a := arcsin(1/(2*sqrt(t^3))) (the principal value of arc sine is considered here).
The Perrin polynomial is the characteristic polynomial of the Perrin recurrence sequence (see A001608):
A(n) = A(n-2) + A(n-3), with A(0)=3, A(1)=0, and A(2)=2.
The Binet formula of this sequence has the form
A(n) = t^(-n) + i^n * t^(n/2) * (e^(i*(a + Pi)*n) + e^(-i*a*n)) = t^(-n) + 2*(-1)^n*t^(n/2)*cos((a + Pi/2)*n),
which implies the relations
A(2*n) = t^(-2*n) + 2 * (-1)^n * cos(2*a*n) * t^n, and
A(2*n-1) = t^(-2*n+1) + 2 * (-1)^(n-1) * sin((2*n-1)*a) * t^(n - 1/2).
It is proved in the paper of Witula et al. that we have
u + v + w = 0 for the respective complex values of the roots: u in (1 + t^(-1))^(1/3), v in (1 + i*sqrt(t)*e^(i*a))^(1/3) and w in (1 - i*sqrt(t)*e^(-i*a))^(1/3).

Examples

			0.8669386054934201...

References

  • R. Witula, E. Hetmaniok, and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, submitted to Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

Crossrefs

Cf. A075778, A060006, A001608.

Programs

  • Mathematica
    ArcSin[1/(2*Root[Function[x, x^3+x^2-1], 1]^(3/2))] // RealDigits[#, 10, 120]& // First (* Jean-François Alcover, Feb 20 2014 *)
  • PARI
    asin(1/2/real(polroots(x^3+x^2-1)[1])^1.5) \\ Charles R Greathouse IV, Dec 11 2013

Formula

Equals arccos((1-A060006)/2)/2. - Gerry Martens, Apr 16 2024

Extensions

a(119) corrected by Sean A. Irvine, Apr 16 2024

A191909 Decimal expansion of the limit of the square root of the ratio of consecutive Padovan numbers.

Original entry on oeis.org

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

Views

Author

Fabrice Auzanneau, Jun 19 2011

Keywords

Comments

This is the square root of the inverse of the plastic number A060006: 1.32471795724...
This is the positive root of x^6 + x^4 - 1 = 0 and the square root of A075778.
An algebraic integer of degree 6 and minimal polynomial x^6 + x^4 - 1. - Charles R Greathouse IV, Apr 21 2016

Examples

			0.868836961832709301806569964191...

Links

Crossrefs

Cf. A000931, A060006.

Programs

  • Mathematica
    RealDigits[x/.FindRoot[x^6+x^4==1,{x,.8},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jan 17 2014 *)
  • PARI
    polrootsreal(x^6+x^4-1)[2] \\ Charles R Greathouse IV, Apr 21 2016
Previous Showing 21-24 of 24 results.

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