A006273 Numerators of a continued fraction for (3+sqrt(13))/2. (Formerly M2879)

3, 10, 1297, 2186871697, 10458512317535240383929505297

Offset

0,1

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..4.

Jeffrey Shallit, Predictable regular continued cotangent expansions, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.

Formula

From Peter Bala, Jan 19 2022: (Start)

a(n) = (11/2 + 3/2*sqrt(13))^3^(n-1) + (11/2 - 3/2*sqrt(13))^3^(n-1) - 1.

a(1) = 10 and a(n) = a(n-1)^3 + 3*a(n-1)^2 - 3 for n >= 2.

a(1) = 10 and a(n) = 13*(Product_{k = 1..n-1} a(k))^2 - 3 for n >= 2.

a(n) = A006268(n-1)^2 + 1 for n >= 1.

13 - 9*Product_{n = 1..N} (1 + 2/a(n))^2 = 52/(a(N+1) + 3). Therefore

sqrt(13) = 3*(1 + 2/10) * (1 + 2/1297) * (1 + 2/2186871697) * ... The convergence is cubic: the first six factors of the product give sqrt(13) correct to more than 750 decimal places.

3/sqrt(13) = (1 - 2/(10+2)) * (1 - 2/(1297+2)) * (1 - 2/(2186871697+2)) * .... (End)

Maple

a := proc (n) option remember; if n = 1 then 10 else a(n-1)^3 + 3*a(n-1)^2 - 3 end if; end proc:
seq(a(n), n = 1..5); # Peter Bala, Jan 19 2022

Crossrefs

For denominators see A006274.

Cf. A098316, A002814, A006268, A006271, A006275, A006276.

Sequence in context: A349894 A103156 A202712 * A183130 A117526 A051498

Adjacent sequences: A006270 A006271 A006272 * A006274 A006275 A006276

Keyword

nonn,frac

Author

N. J. A. Sloane.

Status

approved