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A cubic quartic recurrence.
0

%I #9 Jul 11 2015 10:46:22

%S 1,1,2,9,745,413500186,70701255783138724397185481,

%T 353412074392865080823440901423426679423573814794711467360597541360306163522857

%N A cubic quartic recurrence.

%C 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.

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

%F a(n) ~ c^(3^n), where c = 1.085072477219577474852112080874481159102040272323161792230192441384737595241... . - _Vaclav Kotesovec_, Dec 18 2014

%e a(2) = a(1)^3 + a(0)^4 = 1^3 + 1^4 = 2.

%e a(3) = a(2)^3 + a(1)^4 = 2^3 + 1^4 = 9.

%e a(4) = a(3)^3 + a(2)^4 = 9^3 + 2^4 = 745.

%e a(5) = a(4)^3 + a(3)^4 = 745^3 + 9^4 = 413500186.

%e a(6) = a(5)^2 + a(4)^4 = 413500186^3 + 745^4 = 70701255783138724397185481.

%t 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 *)

%Y Cf. A000283, A112961, A112969, A114793.

%K easy,nonn

%O 0,3

%A _Jonathan Vos Post_, Feb 21 2006

%E Formula corrected by _Vaclav Kotesovec_, Dec 18 2014