A114957 A 4/3-power Fibonacci sequence.

1, 1, 2, 4, 9, 26, 96, 517, 4589, 80409, 3546873, 544383737, 445042712531, 3398279290987133, 510914600201184438040, 4084427005585662985398294639, 6528922582874884079540382952631569851, 12202683821888699966029264978793346242448495941305

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = ceiling(a(n-1)^(4/3) + a(n-2)^(4/3)) for n > 1 with a(0) = a(1) = 1.

Example

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.

Mathematica

Nest[Append[#, Ceiling[Total[Take[#, -2]^(4/3)]]]&, {1, 1}, 17]  (* Harvey P. Dale, Apr 21 2011 *)

Crossrefs

Cf. A000283, A112961, A112969, A114793.

Sequence in context: A243568 A087378 A004252 * A002773 A112706 A359502

Adjacent sequences: A114954 A114955 A114956 * A114958 A114959 A114960

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 21 2006

Extensions

Corrected and extended by Harvey P. Dale, Apr 21 2011

Comments edited by Petros Hadjicostas, Nov 03 2019

Status

approved