A084822 Signature sequence of x, where x=0.577284608955710564894... (A084823) is the unique number between 0 and 1 having the property that the signature sequence of x is equal to the continued fraction expansion of x.

1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 5, 1, 4, 3, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3, 6, 2, 9, 5, 1, 8, 4, 7, 3, 6, 2, 9, 5, 1, 8, 4, 7, 3, 10, 6, 2, 9, 5, 1, 8, 4, 7, 3, 10, 6, 2, 9, 5, 1, 8, 4, 11, 7, 3

Offset: 1

Author

Paul D. Hanna, Jun 04 2003

Keywords

• cofr
• nonn

Comments

The initial terms are close to those of A023116 since x is close to 1/sqrt(3)=A020760. (See A023116.) - R. J. Mathar, Sep 17 2008

Where do they first differ? - N. J. A. Sloane, Jan 20 2023

The first difference is a(2793) = 1 and A023116(2793) = 57. - Paul D. Hanna and Michael S. Branicky, Jan 21 2023

Examples

			Given x = 0.5772846089557105648944851585150204938530... (A084823), the continued fraction of x equals the signature sequence of x.
To obtain the signature sequence of x, arrange the numbers i+j*x (i,j >= 1) in increasing order like so:
[1+1*x, 1+2*x, 2+1*x, 1+3*x, 2+2*x, 1+4*x, 3+1*x, 2+3*x, 1+5*x, 3+2*x, 2+4*x, 1+6*x, 4+1*x, 3+3*x, 2+5*x, 1+7*x, 4+2*x, 3+4*x, 2+6*x, 5+1*x, 1+8*x, 4+3*x, 3+5*x, 2+7*x, 5+2*x, 1+9*x, 4+4*x, 3+6*x, 6+1*x, 2+8*x, 5+3*x, 1+10*x, 4+5*x, 3+7*x, 6+2*x, 2+9*x, 5+4*x, 1+11*x, 4+6*x, 7+1*x, 3+8*x, 6+3*x, 2+10*x, 5+5*x, 1+12*x, 4+7*x, 7+2*x, 3+9*x, 6+4*x, 2+11*x, ...];
then the sequence of i's is the signature of x, and forms this sequence.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..2000
• Index entries for sequences related to signature sequences

Crossrefs

Cf. A084823 (decimal expansion).

See also A023116.