cp's OEIS Frontend

A119810 Partial quotients of the continued fraction of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.

%I A119810 #15 May 30 2019 04:50:35
%S A119810 2,3,10,132,131104,2199023259648,633825300114114700748888473600,
%T A119810 883423532389192164791648750371459257913741948437810659652423818057613312
%N A119810 Partial quotients of the continued fraction of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.
%C A119810 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,...].
%H A119810 W. W. Adams and J. L. Davison, <a href="https://doi.org/10.1090/S0002-9939-1977-0441879-4">A remarkable class of continued fractions</a>, Proc. Amer. Math. Soc. 65 (1977), 194-198.
%H A119810 P. G. Anderson, T. C. Brown, P. J.-S. Shiue, <a href="http://people.math.sfu.ca/~vjungic/tbrown/tom-28.pdf">A simple proof of a remarkable continued fraction identity</a>, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
%H A119810 J. L. Davison, <a href="https://dx.doi.org/10.1090/S0002-9939-1977-0429778-5">A series and its associated continued fraction</a>, Proc. Amer. Math. Soc., 63 (1977), 29-32.
%F A119810 a(n) = 2^A001333(n-1) + 2^A000129(n-2) for n>1, with a(1)=2.
%e A119810 c = 2.32258852258806773012144068278798408011950250800432925665718...
%e A119810 The partial quotients start:
%e A119810 a(1) = 2^1; a(2) = 2^1 + 2^0; a(3) = 2^3 + 2^1;
%e A119810 a(4) = 2^7 + 2^2; a(5) = 2^17 + 2^5; a(6) = 2^41 + 2^12;
%e A119810 and continue as a(n) = 2^A001333(n-1) + 2^A000129(n-2) where
%e A119810 A001333(n) = ( (1+sqrt(2))^n + (1-sqrt(2))^n )/2;
%e A119810 A000129(n) = ( (1+sqrt(2))^n - (1-sqrt(2))^n )/(2*sqrt(2)).
%t A119810 (* b = A001333 *) b[0] = 1; b[1] = 1; b[n_] := b[n] = 2 b[n-1] + b[n-2]; a[1] = 2; a[n_] := 2^b[n-1] + 2^Fibonacci[n-2, 2]; Array[a, 10] (* _Jean-François Alcover_, May 04 2017 *)
%o A119810 (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))))}
%Y A119810 Cf. A119809 (decimal expansion), A119811 (convergents); A119812 (dual constant).
%K A119810 cofr,nonn
%O A119810 1,1
%A A119810 _Paul D. Hanna_, May 26 2006