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a(n) = floor( ((a(n-1) + 1) / 2)^2 ) + 1 for n >= 1, with a(0) = 0.
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%I #20 Jan 01 2024 19:49:17

%S 0,1,2,3,5,10,31,257,16642,69247363,1198799355237125,

%T 359279973529237254190922184970,

%U 32270524844792355518177347536627638351478874995525184567711

%N a(n) = floor( ((a(n-1) + 1) / 2)^2 ) + 1 for n >= 1, with a(0) = 0.

%C Iterated application of A033638, with a shift.

%H Andrew Howroyd, <a href="/A363257/b363257.txt">Table of n, a(n) for n = 0..16</a>

%H A. V. Aho and N. J. A. Sloane, <a href="http://neilsloane.com/doc/doubly.html">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.

%F a(n) = A033638(a(n-1)+1) for n > 0.

%F log a(n) ~ C * 2^n for some constant C.

%o (PARI) a(n) = if(n < 1, 0, floor( ((a(n-1) + 1) / 2)^2 ) + 1) \\ _Andrew Howroyd_, Jan 01 2024

%Y Cf. A003095, A006894, A014980, A033638.

%K nonn

%O 0,3

%A _Harry Richman_, May 23 2023