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A006273
Numerators of a continued fraction for (3+sqrt(13))/2.
(Formerly M2879)
3
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
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.
Sequence in context: A349894 A103156 A202712 * A183130 A117526 A051498
KEYWORD
nonn,frac
AUTHOR
STATUS
approved