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

3, 10, 1297, 2186871697, 10458512317535240383929505297

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac

References

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

Links

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

Crossrefs

For denominators see A006274.

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

Programs

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

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)