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%I #14 Nov 03 2019 22:39:07
%S 1,1,2,4,9,26,96,517,4589,80409,3546873,544383737,445042712531,
%T 3398279290987133,510914600201184438040,4084427005585662985398294639,
%U 6528922582874884079540382952631569851,12202683821888699966029264978793346242448495941305
%N A 4/3-power Fibonacci sequence.
%C This sequence is related to the following sequences:
%C A112961, "a cubic Fibonacci sequence," a(n) = a(n-1)^3 + a(n-2)^3 for n > 2 with a(1) = a(2) = 1; and
%C A112969, "a quartic Fibonacci sequence," a(n) = a(n-1)^4 + a(n-2)^4 for n > 2 with a(1) = a(2) = 1 (which is the quartic (or biquadratic) analog of the Fibonacci sequence similarly to A000283 being the quadratic analog of the Fibonacci sequence).
%C Primes in this sequence include a(2) = 2. Semiprimes in this sequence include a(n) for n = 3, 4, 5, 7, 8.
%F a(n) = ceiling(a(n-1)^(4/3) + a(n-2)^(4/3)) for n > 1 with a(0) = a(1) = 1.
%e a(2) = ceiling(a(0)^(4/3) + a(1)^(4/3)) = ceiling(1^(4/3) + 1^(4/3)) = 2.
%e a(3) = ceiling(a(1)^(4/3) + a(2)^(4/3)) = ceiling)1^(4/3) + 2^(4/3)) = ceiling(3.5198421) = 4.
%e a(4) = ceiling(2^(4/3) + 4^(4/3)) = ceiling(8.86944631) = 9.
%e a(5) = ceiling(4^(4/3) + 9^(4/3)) = ceiling(25.0703586) = 26.
%e a(6) = ceiling(9^(4/3) + 26^(4/3)) = ceiling(95.7456522) = 96.
%e a(7) = ceiling(26^(4/3) + 96^(4/3)) = ceiling(516.595167) = 517.
%e a(8) = ceiling(96^(4/3) + 517^(4/3)) = ceiling(4588.99022) = 4589.
%t Nest[Append[#,Ceiling[Total[Take[#,-2]^(4/3)]]]&,{1,1},17] (* _Harvey P. Dale_, Apr 21 2011 *)
%Y Cf. A000283, A112961, A112969, A114793.
%K easy,nonn
%O 0,3
%A _Jonathan Vos Post_, Feb 21 2006
%E Corrected and extended by _Harvey P. Dale_, Apr 21 2011
%E Comments edited by _Petros Hadjicostas_, Nov 03 2019
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