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 1-2 of 2 results.

A060755 a(n) = log_2(A060723(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 3, 0, 4, 4, 5, 3, 6, 6, 7, 3, 8, 8, 9, 7, 10, 10, 11, 8, 12, 12, 13, 11, 14, 14, 15, 10, 16, 16, 17, 15, 18, 18, 19, 16, 20, 20, 21, 19, 22, 22, 23, 19, 24, 24, 25, 23, 26, 26, 27, 24, 28, 28, 29, 27, 30, 30, 31, 25, 32, 32, 33, 31, 34, 34, 35, 32
Offset: 0

Views

Author

Vladeta Jovovic, Apr 24 2001

Keywords

Crossrefs

Cf. A060723, A305491.

Programs

  • Mathematica
    Table[IntegerExponent[Denominator[Simplify[((1/2 (Sqrt[3] + 1))^x - (1/2 (Sqrt[3] - 1))^x Cos[Pi x])/Sqrt[3]]], 2], {x, 0, 72}]  (* Peter Luschny, Jun 02 2018 *)

Extensions

Data corrected by Peter Luschny, Jun 02 2018

A305491 a(n) = numerator(r(n)) where r(n) = (((1/2)*(sqrt(3) + 1))^n - ((1/2)*(sqrt(3) - 1))^n * cos(Pi*n))/sqrt(3).

Original entry on oeis.org

0, 1, 1, 3, 2, 11, 15, 41, 7, 153, 209, 571, 195, 2131, 2911, 7953, 679, 29681, 40545, 110771, 37829, 413403, 564719, 1542841, 263445, 5757961, 7865521, 21489003, 7338631, 80198051, 109552575, 299303201, 12776743, 1117014753, 1525870529, 4168755811, 1423656585
Offset: 0

Views

Author

Peter Luschny, Jun 02 2018

Keywords

Comments

Let f(x, y) = ((y+1)^x - (y-1)^x * cos(Pi*x))/(y * 2^x). Then f(n, sqrt(3)) are the rational numbers a(n)/A060723(n) and f(n, sqrt(5)) the Fibonacci numbers A000045(n).
From Paul Curtz, Dec 05 2018: (Start)
The binomial inverse of the rational sequence r(n) starts 0, 1, -1, 3/2, -2, 11/4, -15/4, 41/8, -7, 153/16, -209/16, ... and is up to signs equal to r(n). The difference table starts:
0, 1, 1, 3/2, 2, 11/4, 15/4, 41/8, ...
1, 0, 1/2, 1/2, 3/4, 1, 11/8, 15/8, ...
-1, 1/2, 0, 1/4, 1/4, 3/8, 1/2, 11/16, ...
3/2, -1/2, 1/4, 0, 1/8, 1/8, 3/16, 1/4, ...
...
Let s(n) = 2*r(n+1) - r(n) then s(n) = 1, 2, 5/2, 7/2, 19/4, 13/2, ... = A173299(n)/A173300(n) for n >= 1. (End)

Crossrefs

Cf. A060723 (denominators), A060755, A000045, A305492.
Cf. A173299/A173300.

Programs

  • Mathematica
    Table[Numerator[Simplify[((1/2 (Sqrt[3] + 1))^x - (1/2 (Sqrt[3] - 1))^x Cos[Pi  x])/Sqrt[3]]], {x, 0, 36}]

Formula

A recurrence for r(n) is given in A060723.
Showing 1-2 of 2 results.

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