1,1

Convergents A119811: [2/1,7/3,72/31,9511/4095,1246930216/536870911,...], where the denominators of the convergents are equal to [2^A000129(n-1)-1] and A000129 is the Pell numbers. The number of digits in these partial quotients are (beginning at n=1): [1,1,2,3,6,13,30,72,174,420,1013,2445,5901,14246,34391,83027,...].

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

c = 2.32258852258806773012144068278798408011950250800432925665718...

The partial quotients start:

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

a(4) = 2^7 + 2^2; a(5) = 2^17 + 2^5; a(6) = 2^41 + 2^12;

and continue as a(n) = 2^A001333(n-1) + 2^A000129(n-2) where

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

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

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

Cf. A119809 (decimal expansion), A119811 (convergents); A119812 (dual constant).

Sequence in context: A076927 A064647 A062006 this_sequence A055708 A162647 A132536

Adjacent sequences: A119807 A119808 A119809 this_sequence A119811 A119812 A119813

cofr,nonn

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