Properties of a ouadratic fibonacci recurrence

W. Duke, Stephen J. Greenfield, Eugene Speer

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Abstract

The terms of A000278, the sequence defined by h 0=0, hp 1, and h n+2=h n+1+h n 2, count the trees in certain recursively defined forests. We show that for n large, h n is approximately A sqrt(2)n for n even and h n is approximately B sqrt(2)n for n odd, with A,B > 1 and A not equal to B, and we give estimates of A and B: A is 1.436 ±.001 and B is 1.452 ±.001. The doubly exponential growth of the sequence is not surprising (see, for example, [AS]) but the dependence of the growth on the parity of the subscript is more interesting. Numerical and analytical investigation of similar sequences suggests a possible generalization of this result to a large class of such recursions.

Original languageEnglish (US)
JournalJournal of Integer Sequences
Volume1
StatePublished - Dec 1 1998

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics

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    Duke, W., Greenfield, S. J., & Speer, E. (1998). Properties of a ouadratic fibonacci recurrence. Journal of Integer Sequences, 1.