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%I #12 Jul 11 2015 10:49:28
%S 1,1,2,3,4,6,7,9,10,11,12,13,14,15,15,16,16,16,16,16,16,16,16,16,16,
%T 16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,
%U 16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16
%N a(0) = a(1) = 1, for n>1 a(n) = ceiling(a(n-1)^(3/4) + a(n-2)^(3/4)).
%C A 3/4-power Fibonacci sequence.
%C a(17) = 16 is exactly [16^(3/4) + 16^(3/4) = 16. This is a fixed point, so a(n) = 16 for all n>14. This sequence is related to: A112961 "a cubic Fibonacci sequence" a(n) = a(n-1)^3 + a(n-2)^3, A112969 "a quartic Fibonacci sequence" a(n) = a(n-1)^4 + a(n-2)^4, just as A000283 is the quadratic analog of the Fibonacci sequence.
%e a(2) = ceiling(a(0)^(3/4) + a(1)^(3/4)) = ceiling(1^(3/4) + 1^(3/4)) = 2.
%e a(3) = ceiling(a(1)^(3/4) + a(2)^(3/4)) = ceiling(1^(3/4) + 2^(3/4)) = ceiling(2.68179283) = 3.
%e a(4) = ceiling(2^(3/4) + 3^(3/4)) = ceiling(3.96129989) = 4.
%e a(5) = ceiling(3^(3/4) + 4^(3/4)) = ceiling(5.10793418) = 6.
%e a(6) = ceiling(4^(3/4) + 6^(3/4)) = ceiling(6.66208575) = 7.
%t RecurrenceTable[{a[0]==1,a[1]==1,a[n]==Ceiling[a[n-1]^(3/4)+ a[n-2]^(3/4)]}, a[n],{n,80}] (* _Harvey P. Dale_, Jul 22 2011 *)
%Y Cf. A000283, A112961, A112969, A114793.
%K easy,nonn
%O 0,3
%A _Jonathan Vos Post_, Feb 21 2006
%E Edited by _N. J. A. Sloane_, May 20 2006
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