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A114957
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A 4/3-power Fibonacci sequence.
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0
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1, 1, 2, 4, 9, 26, 96, 517, 4589, 80409, 3546873, 544383737, 445042712531, 3398279290987133, 510914600201184438040, 4084427005585662985398294639, 6528922582874884079540382952631569851, 12202683821888699966029264978793346242448495941305
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OFFSET
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0,3
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COMMENTS
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This sequence is related to the following sequences:
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
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).
Primes in this sequence include a(2) = 2. Semiprimes in this sequence include a(n) for n = 3, 4, 5, 7, 8.
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LINKS
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FORMULA
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a(n) = ceiling(a(n-1)^(4/3) + a(n-2)^(4/3)) for n > 1 with a(0) = a(1) = 1.
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EXAMPLE
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a(2) = ceiling(a(0)^(4/3) + a(1)^(4/3)) = ceiling(1^(4/3) + 1^(4/3)) = 2.
a(3) = ceiling(a(1)^(4/3) + a(2)^(4/3)) = ceiling)1^(4/3) + 2^(4/3)) = ceiling(3.5198421) = 4.
a(4) = ceiling(2^(4/3) + 4^(4/3)) = ceiling(8.86944631) = 9.
a(5) = ceiling(4^(4/3) + 9^(4/3)) = ceiling(25.0703586) = 26.
a(6) = ceiling(9^(4/3) + 26^(4/3)) = ceiling(95.7456522) = 96.
a(7) = ceiling(26^(4/3) + 96^(4/3)) = ceiling(516.595167) = 517.
a(8) = ceiling(96^(4/3) + 517^(4/3)) = ceiling(4588.99022) = 4589.
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MATHEMATICA
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Nest[Append[#, Ceiling[Total[Take[#, -2]^(4/3)]]]&, {1, 1}, 17] (* Harvey P. Dale, Apr 21 2011 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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