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

1,3

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..2000

Index entries for sequences related to signature sequences

Example

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.

Crossrefs

Cf. A084823 (decimal expansion).

See also A023116.

Sequence in context: A196059 A272900 A023116 * A292224 A023130 A084532

Adjacent sequences: A084819 A084820 A084821 * A084823 A084824 A084825

Keyword

cofr,nonn

Author

Paul D. Hanna, Jun 04 2003

Status

approved