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

Author

Jason Earls, Jul 24 2001

Keywords

• easy
• nonn

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

Examples

			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.
		

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, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.

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.

Programs

Mathematica

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

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

Formula

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

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