1,3

Convergents begin: [0/1, 1/1, 6/7, 109/127, 112494/131071,...], where the denominators of the convergents are equal to [2^A001333(n-1)-1], where A001333 are numerators of continued fraction convergents to sqrt(2). The number of digits in these partial quotients are (beginning at n=2): [1,1,2,4,8,18,43,102,246,594,1432,3457,8345,20146,48636,117417,...].

a(n) = 4^A000129(n-2) + 2^A001333(n-3) for n>2, with a(1)=0, a(2)=1.

c = 0.858267656461002055792260308433375148664905190083506778667684867..

The partial quotients start:

a(1) = 0; a(2) = 1; a(3) = 4^1 + 2^1; a(4) = 4^2 + 2^1;

a(5) = 4^5 + 2^3; a(6) = 4^12 + 2^7; a(7) = 4^29 + 2^17;

and continue as a(n) = 4^A000129(n-2) + 2^A001333(n-3) where

A000129(n) = ( (1+sqrt(2))^n - (1-sqrt(2))^n )/(2*sqrt(2));

A001333(n) = ( (1+sqrt(2))^n + (1-sqrt(2))^n )/2.

(PARI) {a(n)=if(n==1, 0, if(n==2, 1, 4^round(((1+sqrt(2))^(n-2)+(1-sqrt(2))^(n-2))/(2*sqrt(2))) +if(n==3, 2, 2^round(((1+sqrt(2))^(n-3)-(1-sqrt(2))^(n-3))/2))))}

Cf. A119812 (constant), A119814 (convergents); A119809 (dual constant).

Sequence in context: A052139 A052682 A130437 this_sequence A119986 A130904 A012589

Adjacent sequences: A119810 A119811 A119812 this_sequence A119814 A119815 A119816

cofr,nonn

Paul D. Hanna (pauldhanna(AT)juno.com), May 26 2006