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A076737
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Let u(1)=u(2)=u(3)=2, u(n)=(1+u(n-1)u(n-2))/u(n-3); then a(n) is the numerator of u(n).
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2
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2, 2, 2, 5, 3, 17, 11, 65, 43, 257, 171, 1025, 683, 4097, 2731, 16385, 10923, 65537, 43691, 262145, 174763, 1048577, 699051, 4194305, 2796203, 16777217, 11184811, 67108865, 44739243, 268435457, 178956971, 1073741825, 715827883, 4294967297
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OFFSET
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1,1
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LINKS
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Robert Israel, Table of n, a(n) for n = 1..1000
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FORMULA
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For n>4, a(n)=2^A028242(n-4)*u(n); u(2n)=2^(n-1)+1/2^n hence a(2n)=4^(n-1)+1.
a(1)=a(2)=a(3)=2, a(n+2)=(1+2^n)/(1+2*(n mod 2)). For k>=2, a(2k+1)=A001045(2k-1). - Michael Somos (via Benoit Cloitre), Nov 29 2002
Empirical g.f.: x*(4*x^6+x^4-5*x^3-8*x^2+2*x+2) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)). - Colin Barker, Oct 14 2014
This follows from the Somos formula for a(n+2). - Robert Israel, Aug 10 2015
a(1)=a(2)=a(3)=2 and, for n>3, a(n) = denominator(1/2+6/(4+2^n)). - Gerry Martens, Aug 10 2015
a(n) = H(n - 2, n mod 2, 1/2) for n >= 5 where H(n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], -8). - Peter Luschny, Sep 03 2019
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MAPLE
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2, 2, 2, seq(2/3+(1/6)*2^k+(1/12)*(-1)^k*2^k+(1/3)*(-1)^k, k=4..50); # Robert Israel, Aug 10 2015
H := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], -8):
a := n -> `if`(n < 5, [2, 2, 2, 5][n], H(n-2, irem(n, 2), 1/2)):
seq(simplify(a(n)), n=1..34); # Peter Luschny, Sep 03 2019
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CROSSREFS
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Cf. A005246, A076736 (denominator of u(n)).
Sequence in context: A333388 A174577 A194684 * A246119 A210562 A208512
Adjacent sequences: A076734 A076735 A076736 * A076738 A076739 A076740
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KEYWORD
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nonn,frac
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AUTHOR
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Benoit Cloitre, Nov 24 2002
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STATUS
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approved
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