

A114955


A 2/3power Fibonacci sequence.


0



1, 1, 2, 3, 4, 5, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8
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OFFSET

0,3


COMMENTS

a(11) = 8 is exact [8^(2/3) + 8^(2/3) = 8.00000000]. It is also a fixed point of this sum of 2/3power mapping, so that a(n) = 8 for all n>8. This sequence is related to: A112961 "a cubic Fibonacci sequence" a(1) = a(2) = 1; for n>2: a(n) = a(n1)^3 + a(n2)^3 A112969 "a quartic Fibonacci sequence" a(1) = a(2) = 1; for n>2: a(n) = a(n1)^4 + a(n2)^4, 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(n) for n = 2, 3, 5, 7, 8. Semiprimes in this sequence include a(n) for n = 4, 6.


LINKS

Table of n, a(n) for n=0..77.


FORMULA

a(0) = a(1) = 1, for n>1 a(n) = ceiling(a(n1)^(2/3) + a(n2)^(2/3)).
Euler transform of length 8 sequence [ 1, 1, 1, 0, 0, 1, 0, 1].  Michael Somos, Aug 31 2006
G.f.: (1x^6)(1x^8)/((1x)(1x^2)(1x^3)).  Michael Somos, Aug 31 2006


EXAMPLE

a(2) = ceiling(a(0)^(2/3) + a(1)^(2/3)) = ceiling(1^(2/3) + 1^(2/3)) = 2.
a(3) = ceiling(a(1)^(2/3) + a(2)^(2/3)) = ceiling(1^(2/3) + 2^(2/3)) = ceiling(2.58740105) = 3.
a(4) = ceiling(2^(2/3) + 3^(2/3)) = ceiling(3.66748488) = 4.
a(5) = ceiling(3^(2/3) + 4^(2/3)) = ceiling(4.59992592) = 5.
a(6) = ceiling(4^(2/3) + 5^(2/3)) = ceiling(5.44385984) = 6.
a(7) = ceiling(5^(2/3) + 6^(2/3)) = ceiling(6.22594499) = 7.
a(8) = ceiling(6^(2/3) + 7^(2/3)) = ceiling(6.96123296) = 7.


MATHEMATICA

nxt[{a_, b_}]:={b, Ceiling[b^(2/3)+a^(2/3)]}; Transpose[NestList[nxt, {1, 1}, 80]][[1]] (* Harvey P. Dale, Jan 03 2013 *)


PROG

(PARI) {a(n)=if(n<1, n==0, if(n>8, 8, n(n>7)))} /* Michael Somos, Aug 31 2006 */


CROSSREFS

Cf. A000283, A112961, A112969, A114793.
Sequence in context: A132125 A102672 A287643 * A209384 A060207 A195932
Adjacent sequences: A114952 A114953 A114954 * A114956 A114957 A114958


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Feb 21 2006


STATUS

approved



