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A076725
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a(n)=a(n-1)^2+a(n-2)^4, a(0)=a(1)=1.
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3
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OFFSET
| 0,3
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COMMENTS
| a(n) and a(n+1) are relatively prime for n>=0.
The number of independent sets on a complete binary tree with 2^(n-1)-1 nodes. - Jonathan S. Braunhut (jonbraunhut(AT)usa.net), May 04 2004. For example, when n=3, the complete binary tree with 2 levels has 2^2-1 nodes and has 5 independent sets so a(3)=5. The recursion for number of independent sets splits in two cases, with or without the root node being in the set.
a(6) has 113 digits and is too large to include. This sequence is related to A112969 "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) analogue of the Fibonacci sequence just as A000283 is the quadratic analogue of the Fibonacci sequence. Primes in this sequence include a(n) for n = 2, 3, 4. Semiprimes in this sequence include a(n) for n = 5, 6, 7. - Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 21 2006
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LINKS
| Index entries for sequences of form a(n+1)=a(n)^2 + ...
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FORMULA
| If b(n)=1+1/b(n-1)^2, b(1)=1, then b(n)=a(n)/a(n-1)^2.
a(n)=a(n-1)^2+a(n-2)^4, a(0)=a(1)=1.
Limit a(n)/a(n-1)^2 = A092526 constant. a(n) asymptotic to c1^(2^n)*c2.
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EXAMPLE
| a(2) = a(1)^2 + a(0)^4 = 1^2 + 1^4 = 2.
a(3) = a(2)^2 + a(1)^4 = 2^2 + 1^4 = 5.
a(4) = a(3)^2 + a(2)^4 = 5^2 + 2^4 = 41.
a(5) = a(4)^2 + a(3)^4 = 41^2 + 5^4 = 2306.
a(6) = a(5)^2 + a(4)^4 = 2306^2 + 41^4 = 8143397.
a(7) = a(6)^2 + a(5)^4 = 8143397^2 + 2306^4 = 94592167328105.
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MATHEMATICA
| f[n_]:=(n+1/n)/n; Prepend[Prepend[Numerator[NestList[f, 2, 8]], 1], 1] [From Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), 2010Nov19]
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PROG
| (PARI) a(n)=if(n<2, n>=0, a(n-1)^2+a(n-2)^4)
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CROSSREFS
| Cf. A000283, A112969, A114793.
Sequence in context: A175172 A126469 A054859 * A059917 A093625 A042447
Adjacent sequences: A076722 A076723 A076724 * A076726 A076727 A076728
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KEYWORD
| nonn
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AUTHOR
| Michael Somos Oct 29 2002.
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Andrew Plewe, Jun 15 2007
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