A114954 A 3/2-power Fibonacci sequence.

1, 1, 2, 4, 11, 45, 339, 6544, 535619, 392527477, 7777266564708, 21689055127418446258, 101009204076980364695686091211

Offset: 0

Author

Jonathan Vos Post, Feb 21 2006

Keywords

• easy
• nonn

Comments

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, 4. Semiprimes in this sequence include a(n) for n = 3, 6.

Examples

			a(2) = ceiling(a(0)^(3/2) + a(1)^(3/2)) = ceiling(1^1.5 + 1^1.5) = 2.
a(3) = ceiling(a(1)^(3/2) + a(2)^(3/2)) = ceiling(1^1.5 + 2^1.5) = ceiling(3.82842712) = 4.
a(4) = ceiling(2^(3/2) + 4^(3/2)) = ceiling(10.8284271) = 11.
a(5) = ceiling((4^(3/2)) + (11^(3/2))) = ceiling(44.4828727) = 45.
a(6) = ceiling((11^(3/2)) + (45^(3/2))) = ceiling(338.35205) = 339.
a(7) = ceiling((45^(3/2)) + (339^(3/2))) = ceiling(6543.52112) = 6544.
		

Crossrefs

Cf. A000283, A112961, A112969, A114793.

Programs

Mathematica

RecurrenceTable[{a[0]==a[1]==1,a[n]==Ceiling[Surd[ a[n-1]^3,2]+ Surd[ a[n-2]^3, 2]]},a,{n,15}] (* Harvey P. Dale, Apr 07 2016 *)

Formula

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