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A114955 A 2/3-power 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 (list; graph; refs; listen; history; text; internal format)
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 2/3-power 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(n-1)^3 + a(n-2)^3 A112969 "a quartic Fibonacci sequence" a(1) = a(2) = 1; for n>2: a(n) = a(n-1)^4 + a(n-2)^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(n-1)^(2/3) + a(n-2)^(2/3)).

Euler transform of length 8 sequence [ 1, 1, 1, 0, 0, -1, 0, -1]. - Michael Somos, Aug 31 2006

G.f.: (1-x^6)(1-x^8)/((1-x)(1-x^2)(1-x^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

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Last modified July 11 17:24 EDT 2020. Contains 335626 sequences. (Running on oeis4.)