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A063438 a(n) = floor((n+1)*Pi) - floor(n*Pi).

%I A063438 #33 May 27 2025 07:12:03
%S A063438 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 A063438 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 A063438 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 A063438 a(n) = floor((n+1)*Pi) - floor(n*Pi).
%C A063438 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 A063438 A word that is uniformly recurrent, but not morphic. - _N. J. A. Sloane_, Jul 14 2018
%D A063438 G. H. Hardy, Divergent Series, Oxford 1979.
%D A063438 Zeller, K. and Beekmann, W., Theorie der Limitierungsverfahren. Springer Verlag, Berlin, 1970.
%H A063438 Harry J. Smith, <a href="/A063438/b063438.txt">Table of n, a(n) for n = 1..2000</a>
%H A063438 Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, <a href="https://arxiv.org/abs/1711.10807">A Taxonomy of Morphic Sequences</a>, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.
%F A063438 a(n) = A115790(n) + 3. - _Michel Marcus_, Jul 15 2013
%e A063438 a(6)=3 because 7*Pi = 21.99..., 6*Pi = 18.84..., so a(6) = 21 - 18;
%e A063438 a(7)=4 because 8*Pi = 25.13..., 7*Pi = 21.99..., so a(7) = 25 - 21.
%t A063438 Differences[Floor[Pi Range[120]]] (* _Harvey P. Dale_, Jul 02 2021 *)
%o A063438 (PARI) j=[]; for(n=1,150,j=concat(j, floor( (n+1) * Pi) - floor(n * Pi))); j
%o A063438 (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 A063438 (PARI) a(n) = floor((n+1)*Pi) - floor(n*Pi) \\ _Michel Marcus_, Jul 15 2013
%Y A063438 Cf. A000796, A115788, A115789, A115790, A006337.
%Y A063438 First differences of A022844.
%Y A063438 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 A063438 easy,nonn
%O A063438 1,1
%A A063438 _Jason Earls_, Jul 24 2001
%E A063438 Offset in b-file and second PARI program corrected by _N. J. A. Sloane_, Aug 31 2009
%E A063438 Entry revised by _N. J. A. Sloane_, Jan 07 2014