The Pell number is a mathematical sequence of positive whole numbers , the Pell numbers (Engl. Pell numbers ), as well as the Pell numbers second type (Engl. Companion Pell numbers ). It takes its name from the English mathematician John Pell (1611–1685).
Pell sequence / numbers
The sequence is defined recursively by:
That means in words:
- the values zero and one are specified for the first two numbers
- each additional number is calculated by doubling the direct predecessor and then adding the previous one.
The first numbers in the sequence are (if you start counting with ):
- 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, ... (sequence A000129 in OEIS )
The Pell sequence can also be interpreted as a special case of the general Lucas sequence with and :
Silver cut
The following applies to the limit value of the ratio of two consecutive numbers in the Pell sequence:
This number is called the silver ratio in analogy to the golden ratio of the Fibonacci sequence .
Derivation of the numerical value
The following limit value must be determined:
With follows:
With
further follows . This gives the quadratic equation
with the two solutions and .
Since of these two values only the positive one comes into question for the limit value, it follows:
Closed form of the Pell series
In the section Deriving the numerical value , the ratio of two consecutive numbers of the Pell sequence was shown for the limit values:
-
and .
Be and real constants. Then meet the geometric sequences
-
and
the recursion formulas
-
and
-
.
Their linear combination also fulfills the Pell recursion.
The following initial values must apply to the Pell sequence: and .
Inserted in results in the following system of equations:
-
and
with the solutions and
This gives the closed form of the Pell sequence:
Generating function of the Pell sequence
The generating function of the Pell sequence is:
This power series has the radius of convergence .
Derivation of the function
The generating function of the Pell sequence has the radius of convergence .
The following applies to :
Pell primes
A Pell prime is a Pell number that is prime. The smallest Pell prime numbers are:
- 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449, 4760981394323203445293052612223893281, ... (sequence A086383 in OEIS )
For these Pell primes the index is of the following:
- 2, 3, 5, 11, 13, 29, 41, 53, 59, 89, 97, 101, 167, 181, 191, 523, 929, 1217, 1301, 1361, 2087, 2273, 2393, 8093, 13339, 14033, 23747, 28183, 34429, 36749, 90197,… (Follow A096650 in OEIS )
-
Example 1:
- It is and . Thus is a prime number. In fact, the index appears in the above list in 4th position because it leads to the fourth smallest Pell prime number .
The following properties apply to Pell primes:
- If a Pell is prime, then the index is also a prime (the converse is incorrect, meaning that not every prime index leads to a Pell prime).
Pell Numbers 2nd Kind / Companion Pell Series
Pell numbers of the 2nd type are also called Pell-Lucas numbers .
The sequence is defined recursively by:
That means in words:
- the value two is given for the first two numbers
- each additional number is calculated by doubling the direct predecessor and then adding the previous one.
The first numbers in the sequence are 2, 2, 6, 14, 34, 82, 198, 478, 1154, ... (sequence A002203 in OEIS )
The Companion Pell sequence can also be interpreted as a special case of the general Lucas sequence with and :
Web links
Individual evidence
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↑ Comments on OEIS A096650
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formula based
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Carol ((2 n - 1) 2 - 2) |
Cullen ( n ⋅2 n + 1) |
Double Mersenne (2 2 p - 1 - 1) |
Euclid ( p n # + 1) |
Factorial ( n! ± 1) |
Fermat (2 2 n + 1) |
Cubic ( x 3 - y 3 ) / ( x - y ) |
Kynea ((2 n + 1) 2 - 2) |
Leyland ( x y + y x ) |
Mersenne (2 p - 1) |
Mills ( A 3 n ) |
Pierpont (2 u ⋅3 v + 1) |
Primorial ( p n # ± 1) |
Proth ( k ⋅2 n + 1) |
Pythagorean (4 n + 1) |
Quartic ( x 4 + y 4 ) |
Thabit (3⋅2 n - 1) |
Wagstaff ((2 p + 1) / 3) |
Williams (( b-1 ) ⋅ b n - 1)
Woodall ( n ⋅2 n - 1)
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Prime number follow
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Bell |
Fibonacci |
Lucas |
Motzkin |
Pell |
Perrin
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property-based
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Elitist |
Fortunate |
Good |
Happy |
Higgs |
High quotient |
Isolated |
Pillai |
Ramanujan |
Regular |
Strong |
Star |
Wall – Sun – Sun |
Wieferich |
Wilson
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base dependent
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Belphegor |
Champernowne |
Dihedral |
Unique |
Happy |
Keith |
Long |
Minimal |
Mirp |
Permutable |
Primeval |
Palindrome |
Repunit ((10 n - 1) / 9) |
Weak |
Smarandache – Wellin |
Strictly non-palindromic |
Strobogrammatic |
Tetradic |
Trunkable |
circular
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based on tuples
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Balanced ( p - n , p , p + n) |
Chen |
Cousin ( p , p + 4) |
Cunningham ( p , 2 p ± 1, ...) |
Triplet ( p , p + 2 or p + 4, p + 6) |
Constellation |
Sexy ( p , p + 6) |
Safe ( p , ( p - 1) / 2) |
Sophie Germain ( p , 2 p + 1) |
Quadruplets ( p , p + 2, p + 6, p + 8) |
Twin ( p , p + 2) |
Twin bi-chain ( n ± 1, 2 n ± 1, ...)
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according to size
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Titanic (1,000+ digits) |
Gigantic (10,000+ digits) |
Mega (1,000,000+ digits) |
Beva (1,000,000,000+ positions)
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Composed
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Carmichael |
Euler's pseudo |
Almost |
Fermatsche pseudo |
Pseudo |
Semi |
Strong pseudo |
Super Euler's pseudo
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