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.
%I A003231 M2618 #69 Jan 05 2025 19:51:33
%S A003231 3,7,10,14,18,21,25,28,32,36,39,43,47,50,54,57,61,65,68,72,75,79,83,
%T A003231 86,90,94,97,101,104,108,112,115,119,123,126,130,133,137,141,144,148,
%U A003231 151,155,159,162,166,170,173,177,180,184,188,191,195,198,202,206,209
%N A003231 a(n) = floor(n*(sqrt(5)+5)/2).
%C A003231 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 A003231 This is the function named c in the Carlitz-Scoville-Vaughan link. - _Eric M. Schmidt_, Aug 06 2015
%C A003231 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 A003231 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 A003231 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003231 L. Carlitz, R. Scoville and T. Vaughan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/carlitz.pdf">Some arithmetic functions related to Fibonacci numbers</a>, Fib. Quart., 11 (1973), 337-386.
%H A003231 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 A003231 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 A003231 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A003231 a(n) = 2*n + A000201(n). - _R. J. Mathar_, Aug 22 2014
%p A003231 A003231:=n->floor(n*(sqrt(5)+5)/2): seq(A003231(n), n=1..100); # _Wesley Ivan Hurt_, Aug 06 2015
%t A003231 With[{c=(Sqrt[5]+5)/2}, Floor[c*Range[60]]] (* _Harvey P. Dale_, Oct 01 2012 *)
%o A003231 (PARI) a(n)=floor(n*(sqrt(5)+5)/2)
%o A003231 (PARI) a(n)=(5*n+sqrtint(5*n^2))\2; \\ _Michel Marcus_, Nov 14 2023
%o A003231 (Haskell)
%o A003231 a003231 = floor . (/ 2) . (* (sqrt 5 + 5)) . fromIntegral
%o A003231 -- _Reinhard Zumkeller_, Oct 03 2014
%o A003231 (Magma) [Floor(n*(Sqrt(5)+5)/2): n in [1..100]]; // _Vincenzo Librandi_, Oct 23 2014
%o A003231 (Python)
%o A003231 from math import isqrt
%o A003231 def A003231(n): return (n+isqrt(5*n**2)>>1)+(n<<1) # _Chai Wah Wu_, Aug 25 2022
%Y A003231 Cf. A000201, A003231, A003233, A003234, A249115.
%Y A003231 Cf. A105424, A362917.
%K A003231 nonn,easy
%O A003231 1,1
%A A003231 _N. J. A. Sloane_
%E A003231 Better description and more terms from _Michael Somos_, Jun 07 2000