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a(n) = 2^(n-1) + (2 + (-1)^n)^((n-2)/2).
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%I #12 Apr 07 2024 07:13:43

%S 2,3,5,11,17,41,65,155,257,593,1025,2291,4097,8921,16385,34955,65537,

%T 137633,262145,543971,1048577,2156201,4194305,8565755,16777217,

%U 34085873,67108865,135812051,268435457,541653881,1073741825,2161832555,4294967297,8632981313

%N a(n) = 2^(n-1) + (2 + (-1)^n)^((n-2)/2).

%C Extends Euler's 6-term sequence.

%C In Euler's 6-term sequence we have noticed that e(1) = 2 = 2^0 + 1; e(2) = 3 = 2^1 + 3^0; e(3) = 5 = 2^2 + 1; e(4) = 11 = 2^3 + 3^1; e(5) = 17 = 2^4 + 1; e(6) = 41 = 2^5 + 3^2, which of course immediately leads to our formula above. Note: For m>0, we take m^(1/2) to be the unique positive square root of m.

%C a(2^n + 1), n=0,1,2,... are the Fermat numbers.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-8,-3,6).

%o (PARI) a(n) = 2^(n-1) + (2 + (-1)^n)^((n-2)/2) \\ _Charles R Greathouse IV_, Jan 29 2013

%Y Cf. A082605.

%K nonn,easy

%O 1,1

%A Ben de la Rosa & Johan Meyer (MeyerJH.sci(AT)mail.uovs.ac.za), Jul 09 2003

%E More terms from Neven Juric, Apr 10 2008