%I M2618 #67 Nov 14 2023 09:22:39

%S 3,7,10,14,18,21,25,28,32,36,39,43,47,50,54,57,61,65,68,72,75,79,83,

%T 86,90,94,97,101,104,108,112,115,119,123,126,130,133,137,141,144,148,

%U 151,155,159,162,166,170,173,177,180,184,188,191,195,198,202,206,209

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

%C 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

%C This is the function named c in the Carlitz-Scoville-Vaughan link. - _Eric M. Schmidt_, Aug 06 2015

%C 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

%D 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.

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

%H L. Carlitz, R. Scoville and T. Vaughan, <a href="http://www.fq.math.ca/Scanned/11-4/carlitz.pdf">Some arithmetic functions related to Fibonacci numbers</a>, Fib. Quart., 11 (1973), 337-386.

%H Michel Dekking and Ad van Loon, <a href="https://doi.org/10.48550/arXiv.2111.07544">On the representation of the natural numbers by powers of the golden mean</a>, arXiv:2111.07544 [math.NT], 15 Nov 2021.

%H Scott V. Tezlaf, <a href="https://arxiv.org/abs/1806.00331">On ordinal dynamics and the multiplicity of transfinite cardinality</a>, arXiv:1806.00331 [math.NT], 2018. See p. 9.

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

%F a(n) = 2*n + A000201(n). - _R. J. Mathar_, Aug 22 2014

%p A003231:=n->floor(n*(sqrt(5)+5)/2): seq(A003231(n), n=1..100); # _Wesley Ivan Hurt_, Aug 06 2015

%t With[{c=(Sqrt[5]+5)/2}, Floor[c*Range[60]]] (* _Harvey P. Dale_, Oct 01 2012 *)

%o (PARI) a(n)=floor(n*(sqrt(5)+5)/2)

%o (PARI) a(n)=(5*n+sqrtint(5*n^2))\2; \\ _Michel Marcus_, Nov 14 2023

%o (Haskell)

%o a003231 = floor . (/ 2) . (* (sqrt 5 + 5)) . fromIntegral

%o -- _Reinhard Zumkeller_, Oct 03 2014

%o (Magma) [Floor(n*(Sqrt(5)+5)/2): n in [1..100]]; // _Vincenzo Librandi_, Oct 23 2014

%o (Python)

%o from math import isqrt

%o def A003231(n): return (n+isqrt(5*n**2)>>1)+(n<<1) # _Chai Wah Wu_, Aug 25 2022

%Y Cf. A000201, A003231, A003233, A003234, A249115.

%Y Cf. A105424, A362917.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E Better description and more terms from _Michael Somos_, Jun 07 2000