cp's OEIS Frontend

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.

%I A084822 #38 May 27 2024 11:22:40
%S A084822 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,
%T A084822 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,
%U A084822 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
%N 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.
%C A084822 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
%C A084822 Where do they first differ? - _N. J. A. Sloane_, Jan 20 2023
%C A084822 The first difference is a(2793) = 1 and A023116(2793) = 57. - _Paul D. Hanna_ and _Michael S. Branicky_, Jan 21 2023
%H A084822 Paul D. Hanna, <a href="/A084822/b084822.txt">Table of n, a(n) for n = 1..2000</a>
%H A084822 <a href="/index/Si#signature_sequences">Index entries for sequences related to signature sequences</a>
%e A084822 Given x = 0.5772846089557105648944851585150204938530... (A084823), the continued fraction of x equals the signature sequence of x.
%e A084822 To obtain the signature sequence of x, arrange the numbers i+j*x (i,j >= 1) in increasing order like so:
%e A084822 [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, ...];
%e A084822 then the sequence of i's is the signature of x, and forms this sequence.
%Y A084822 Cf. A084823 (decimal expansion).
%Y A084822 See also A023116.
%K A084822 cofr,nonn
%O A084822 1,3
%A A084822 _Paul D. Hanna_, Jun 04 2003