%I #28 Apr 13 2024 12:47:54
%S 3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,
%T 4,3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,
%U 3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,4,3,3,3,3,3,3,4
%N a(n) = floor((n+1)*Pi) - floor(n*Pi).
%C The arithmetic mean (1/(n+1))*Sum_{k=0..n} a(k) converges to Pi. What is effectively the same: the Cesaro limit (C1) of a(n) is Pi. - _Hieronymus Fischer_, Jan 31 2006
%C A word that is uniformly recurrent, but not morphic. - _N. J. A. Sloane_, Jul 14 2018
%D G. H. Hardy, Divergent Series, Oxford 1979.
%D Zeller, K. and Beekmann, W., Theorie der Limitierungsverfahren. Springer Verlag, Berlin, 1970.
%H Harry J. Smith, <a href="/A063438/b063438.txt">Table of n, a(n) for n = 1..2000</a>
%H Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, <a href="https://arxiv.org/pdf/1711.10807.pdf">A Taxonomy of Morphic Sequences</a>, arXiv preprint arXiv:1711.10807, Nov 29 2017.
%F a(n) = A115790(n) + 3. - _Michel Marcus_, Jul 15 2013
%e a(6)=3 because 7*Pi = 21.99..., 6*Pi = 18.84..., so a(6) = 21 - 18;
%e a(7)=4 because 8*Pi = 25.13..., 7*Pi = 21.99..., so a(7) = 25 - 21.
%t Differences[Floor[Pi Range[120]]] (* _Harvey P. Dale_, Jul 02 2021 *)
%o (PARI) j=[]; for(n=1,150,j=concat(j, floor( (n+1) * Pi) - floor(n * Pi))); j
%o (PARI) { default(realprecision, 50); for (n=1, 2000, write("b063438.txt", n, " ", floor((n + 1)*Pi) - floor(n*Pi)) ) } \\ _Harry J. Smith_, Aug 21 2009
%o (PARI) a(n) = floor((n+1)*Pi) - floor(n*Pi) \\ _Michel Marcus_, Jul 15 2013
%Y Cf. A000796, A115788, A115789, A115790, A006337.
%Y First differences of A022844.
%Y Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.
%K easy,nonn
%O 1,1
%A _Jason Earls_, Jul 24 2001
%E Offset in b-file and second PARI program corrected by _N. J. A. Sloane_, Aug 31 2009
%E Entry revised by _N. J. A. Sloane_, Jan 07 2014