A061377 a(1) = 1, a(n+1) = numerator of the continued fraction [1; 2, 4, 8, ..., 2^n].

1, 3, 13, 107, 1725, 55307, 3541373, 453351051, 116061410429, 59423895490699, 60850185043886205, 124621238393774438539, 510448653311085144141949, 4181595492545647894585284747, 68511261060316548415970449436797

Offset: 1

Author

Amarnath Murthy, May 02 2001

Keywords

• nonn
• easy
• frac

Examples

			G.f. = x + 3*x^2 + 13*x^3 + 107*x^4 + 1725*x^5 + 55307*x^6 + 3541373*x^7 + ...
a(3) = 13, the numerator of 1 + 1/(2 + 1/4) = 13/9.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..82

Crossrefs

Denominators are sequence A015473.

Programs

Maple

with(numtheory); f := n->numer(cfrac([seq (2^i,i=0..n)])); for n from 0 to 25 do printf("%d,",f(n)) od;

Mathematica

Module[{nn=20,c},c=2^Range[0,nn];Table[Numerator[ FromContinuedFraction[ Take[ c,n]]],{n,nn}]] (* Harvey P. Dale, Jun 04 2014 *)

PARI

{a(n) = if( n<1, 0, n<3, 2*n-1, 2^(n-1)*a(n-1) + a(n-2))}; /* Michael Somos, Dec 28 2016 */

Formula

a(n) = 2^(n-1)*a(n-1) + a(n-2). - Orson R. L. Peters, Dec 28 2016

0 = a(n)*(-2*a(n+2)) + a(n+1)*(+a(n+1) - a(n+3)) + a(n+2)*(+2*a(n+2)) if n>0. - Michael Somos, Dec 28 2016

Extensions

More terms from Larry Reeves (larryr(AT)acm.org) and Winston C. Yang (winston(AT)cs.wisc.edu), May 15 2001