A114953 A cubic quartic recurrence.

1, 1, 2, 9, 745, 413500186, 70701255783138724397185481, 353412074392865080823440901423426679423573814794711467360597541360306163522857

Offset: 0

Author

Jonathan Vos Post, Feb 21 2006

Keywords

• easy
• nonn

Comments

a(6) has 233 digits. 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. Semiprimes in this sequence include a(n) for n = 3, 4, 6.

Examples

			a(2) = a(1)^3 + a(0)^4 = 1^3 + 1^4 = 2.
a(3) = a(2)^3 + a(1)^4 = 2^3 + 1^4 = 9.
a(4) = a(3)^3 + a(2)^4 = 9^3 + 2^4 = 745.
a(5) = a(4)^3 + a(3)^4 = 745^3 + 9^4 = 413500186.
a(6) = a(5)^2 + a(4)^4 = 413500186^3 + 745^4 = 70701255783138724397185481.
		

Crossrefs

Cf. A000283, A112961, A112969, A114793.

Programs

Mathematica

RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == a[n-1]^3 + a[n-2]^4}, a, {n, 0, 8}] (* Vaclav Kotesovec, Dec 18 2014 *)

Formula

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

a(n) ~ c^(3^n), where c = 1.085072477219577474852112080874481159102040272323161792230192441384737595241... . - Vaclav Kotesovec, Dec 18 2014

Extensions

Formula corrected by Vaclav Kotesovec, Dec 18 2014