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

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, 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, 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

Offset

1,1

Comments

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

A word that is uniformly recurrent, but not morphic. - N. J. A. Sloane, Jul 14 2018

References

G. H. Hardy, Divergent Series, Oxford 1979.

Zeller, K. and Beekmann, W., Theorie der Limitierungsverfahren. Springer Verlag, Berlin, 1970.

Links

Harry J. Smith, Table of n, a(n) for n = 1..2000

Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807, Nov 29 2017.

Formula

a(n) = A115790(n) + 3. - Michel Marcus, Jul 15 2013

Example

a(6)=3 because 7*Pi = 21.99..., 6*Pi = 18.84..., so a(6) = 21 - 18;
a(7)=4 because 8*Pi = 25.13..., 7*Pi = 21.99..., so a(7) = 25 - 21.

Mathematica

Differences[Floor[Pi Range[120]]] (* Harvey P. Dale, Jul 02 2021 *)

Prog

(PARI) j=[]; for(n=1, 150, j=concat(j, floor( (n+1) * Pi) - floor(n * Pi))); j
(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
(PARI) a(n) = floor((n+1)*Pi) - floor(n*Pi) \\ Michel Marcus, Jul 15 2013

Crossrefs

Cf. A000796, A115788, A115789, A115790, A006337.

First differences of A022844.

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.

Sequence in context: A083565 A237117 A247970 * A276870 A081168 A301415

Adjacent sequences: A063435 A063436 A063437 * A063439 A063440 A063441

Keyword

easy,nonn

Author

Jason Earls, Jul 24 2001

Extensions

Offset in b-file and second PARI program corrected by N. J. A. Sloane, Aug 31 2009

Entry revised by N. J. A. Sloane, Jan 07 2014

Status

approved