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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A003258 The number m such that c'(m) = A005206(A003231(n)), where c'(m) = A249115(m) is the m-th positive integer not in A003231.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 71, 73, 75, 76, 78, 80, 81, 83, 84, 86, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 104, 105
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

This is the function named phi in the Carlitz-Scoville-Vaughan link. - Eric M. Schmidt, Aug 14 2014

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Formula

Conjecture: a(n) = A078489(n) + n - 1. - Ralf Stephan, Feb 24 2004

Extensions

More terms and a definition from Eric M. Schmidt, Aug 17 2014
Definition edited by Eric M. Schmidt, Aug 07 2015

A003231 a(n) = floor(n*(sqrt(5)+5)/2).

Original entry on oeis.org

3, 7, 10, 14, 18, 21, 25, 28, 32, 36, 39, 43, 47, 50, 54, 57, 61, 65, 68, 72, 75, 79, 83, 86, 90, 94, 97, 101, 104, 108, 112, 115, 119, 123, 126, 130, 133, 137, 141, 144, 148, 151, 155, 159, 162, 166, 170, 173, 177, 180, 184, 188, 191, 195, 198, 202, 206, 209
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Let r = (5 - sqrt(5))/2 and s = (5 + sqrt(5))/2. Then 1/r + 1/s = 1, so that A249115 and A003231 are a pair of complementary Beatty sequences. Let tau = (1 + sqrt(5))/2, the golden ratio. Let R = {h*tau, h >= 1} and S = {k*(tau - 1), k >= 1}. Then A003231(n) is the position of n*tau in the ordered union of R and S. The position of n*(tau - 1) is A249115(n). - Clark Kimberling, Oct 21 2014
This is the function named c in the Carlitz-Scoville-Vaughan link. - Eric M. Schmidt, Aug 06 2015
Natural numbers whose representation in base phi differs between the Bergmann representation and the "canonical" representation described by Dekking and van Loon. See proposition 3.3 in Dekking, van Loon (2021). - Hugo Pfoertner, May 26 2023

References

  • Dekking, Michel, and Ad van Loon. "On the representation of the natural numbers by powers of the golden mean." arXiv preprint arXiv:2111.07544 (2021); Fib. Quart. 61:2 (May 2023), 105-118.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000201, A003231, A003233, A003234, A249115.
Cf. A105424, A362917.

Programs

  • Haskell
    a003231 = floor . (/ 2) . (* (sqrt 5 + 5)) . fromIntegral
-- Reinhard Zumkeller, Oct 03 2014
  • Magma
    [Floor(n*(Sqrt(5)+5)/2): n  in [1..100]]; // Vincenzo Librandi, Oct 23 2014
  • Maple
    A003231:=n->floor(n*(sqrt(5)+5)/2): seq(A003231(n), n=1..100); # Wesley Ivan Hurt, Aug 06 2015
  • Mathematica
    With[{c=(Sqrt[5]+5)/2}, Floor[c*Range[60]]] (* Harvey P. Dale, Oct 01 2012 *)
  • PARI
    a(n)=floor(n*(sqrt(5)+5)/2)
  • PARI
    a(n)=(5*n+sqrtint(5*n^2))\2; \\ Michel Marcus, Nov 14 2023
  • Python
    from math import isqrt
def A003231(n): return (n+isqrt(5*n**2)>>1)+(n<<1) # Chai Wah Wu, Aug 25 2022

Formula

a(n) = 2*n + A000201(n). - R. J. Mathar, Aug 22 2014

Extensions

Better description and more terms from Michael Somos, Jun 07 2000
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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