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A number in the sequence created from the recurrence relation $$P_n = 2P_{n - 1} + P_{n - 2},$$ with $$P_0 = 0$$ and $$P_1 = 1.$$ The first few Pell numbers are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, listed in A000129 of Sloane's OEIS.
A Pell number for any given index can also be calculated from earlier Pell numbers with $$P_{a + b} = P_aP_{b + 1} + P_{a - 1}P_b.$$
The formula $${ {-{(1 - \sqrt 2)}^n + {(1 + \sqrt 2)}^n} \over {2\sqrt 2}}$$ works too. From this particular formula it can be deduced that the sequence of Pell numbers can be used in a continued fraction of the square root of 2 as well as the silver ratio.
Yet another way to calculate Pell numbers is by squaring the terms of Pascal's triangle and adding up the antidiagonals. Arranging the Markov numbers in a binary graph tree and reading the numbers on 2's branch gives the Pell numbers with odd indices.
Only Pell numbers with prime indexes can also be prime. This fact is used in some tests for pseudoprimality.
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"Pell number" is owned by CompositeFan.
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| Also defines: |
Pell number |
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Cross-references: prime, indices, odd, branch, tree, graph, binary, Markov numbers, Pascal's triangle, terms, calculate, silver ratio, square root of 2, continued fraction, formula, index, OEIS, recurrence relation, sequence, number
There are 3 references to this entry.
This is version 3 of Pell number, born on 2006-03-17, modified 2007-02-16.
Object id is 7736, canonical name is PellNumber.
Accessed 2631 times total.
Classification:
| AMS MSC: | 11B39 (Number theory :: Sequences and sets :: Fibonacci and Lucas numbers and polynomials and generalizations) |
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Pending Errata and Addenda
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