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 A004976 #65 Jun 01 2025 03:34:03
%S A004976 0,4,8,12,16,21,25,29,33,38,42,46,50,55,59,63,67,72,76,80,84,88,93,97,
%T A004976 101,105,110,114,118,122,127,131,135,139,144,148,152,156,160,165,169,
%U A004976 173,177,182,186,190,194,199
%N A004976 a(n) = floor(n*phi^3), where phi=(1+sqrt(5))/2.
%C A004976 For n>=1, a(n) is the position of the n-th 1 in the zero-one sequence [n*r+r]-[n*r]-[r], where r=sqrt(5); see A188221. Also, A004976=-1+A004958 (for n>=1), and A004976 is the complement of A188222. - _Clark Kimberling_, Mar 24 2011
%H A004976 Vincenzo Librandi, <a href="/A004976/b004976.txt">Table of n, a(n) for n = 0..1000</a>
%H A004976 A. J. Hildebrand, Junxian Li, Xiaomin Li, and Yun Xie, <a href="https://arxiv.org/abs/1809.08690">Almost Beatty Partitions</a>, arXiv:1809.08690 [math.NT], 2018.
%H A004976 Vincent Russo and Loren Schwiebert, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/49-2/RussoSchwiebert.pdf">Beatty Sequences, Fibonacci Numbers, and the Golden Ratio</a>, The Fibonacci Quarterly, Vol 49, Number 2, May 2011.
%H A004976 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A004976 a(n) = n+floor(2*n*phi). [Formula corrected by _Clark Kimberling_, Mar 22 2008]
%t A004976 r=5^(1/2); k=1;
%t A004976 t=Table[Floor[n*r+k*r]-Floor[n*r]-Floor[k*r], {n,1,220}] (* A188221 *)
%t A004976 Flatten[Position[t,0] ] (* A188222 *)
%t A004976 Flatten[Position[t,1] ] (* A004976 *)
%t A004976 (* _Clark Kimberling_, Mar 24 2011 *)
%t A004976 With[{c=GoldenRatio^3},Floor[c*Range[0,50]]] (* _Vincenzo Librandi_, Apr 12 2012 *)
%o A004976 (PARI) a(n)=2*n+sqrtint(5*n^2) \\ _Charles R Greathouse IV_, Apr 12 2012
%o A004976 (Python)
%o A004976 from math import isqrt
%o A004976 def A004976(n): return (isqrt(20*n**2)>>1)+(n<<1) # _Chai Wah Wu_, Aug 17 2022
%Y A004976 Cf. A000201, A001950, A004919, A004958, A188221, A188222.
%K A004976 nonn,easy
%O A004976 0,2
%A A004976 _N. J. A. Sloane_