OFFSET
1,1
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
L. Carlitz, R. Scoville and T. Vaughan, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11 (1973), 337-386.
Michel Dekking and Ad van Loon, On the representation of the natural numbers by powers of the golden mean, arXiv:2111.07544 [math.NT], 15 Nov 2021.
Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 9.
FORMULA
a(n) = 2*n + A000201(n). - R. J. Mathar, Aug 22 2014
MAPLE
MATHEMATICA
With[{c=(Sqrt[5]+5)/2}, Floor[c*Range[60]]] (* Harvey P. Dale, Oct 01 2012 *)
PROG
(PARI) a(n)=floor(n*(sqrt(5)+5)/2)
(PARI) a(n)=(5*n+sqrtint(5*n^2))\2; \\ Michel Marcus, Nov 14 2023
(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
(Python)
from math import isqrt
def A003231(n): return (n+isqrt(5*n**2)>>1)+(n<<1) # Chai Wah Wu, Aug 25 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Better description and more terms from Michael Somos, Jun 07 2000
STATUS
approved