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Jacobsthal number

In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence for which P = 1, and Q = −2[1]—and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that. The first Jacobsthal numbers are:

0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … (sequence A001045 in the OEIS)

A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are:

3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, … (sequence A049883 in the OEIS)

Jacobsthal numbers edit

Jacobsthal numbers are defined by the recurrence relation:

 

The next Jacobsthal number is also given by the recursion formula

 

or by

 

The second recursion formula above is also satisfied by the powers of 2.

The Jacobsthal number at a specific point in the sequence may be calculated directly using the closed-form equation:[2]

 

The generating function for the Jacobsthal numbers is

 

The sum of the reciprocals of the Jacobsthal numbers is approximately 2.7186, slightly larger than e.

The Jacobsthal numbers can be extended to negative indices using the recurrence relation or the explicit formula, giving

  (see OEISA077925)

The following identity holds

  (see OEISA139818)

Jacobsthal–Lucas numbers edit

Jacobsthal–Lucas numbers represent the complementary Lucas sequence  . They satisfy the same recurrence relation as Jacobsthal numbers but have different initial values:

 

The following Jacobsthal–Lucas number also satisfies:[2]

 

The Jacobsthal–Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:[2]

 

The first Jacobsthal–Lucas numbers are:

2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, … (sequence A014551 in the OEIS).

Jacobsthal Oblong numbers edit

The first Jacobsthal Oblong numbers are: 0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, … (sequence A084175 in the OEIS)

 

References edit

  1. ^ Weisstein, Eric W. "Jacobsthal Number". MathWorld.
  2. ^ a b c Sloane, N. J. A. (ed.). "Sequence A014551 (Jacobsthal–Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

jacobsthal, number, mathematics, integer, sequence, named, after, german, mathematician, ernst, jacobsthal, like, related, fibonacci, numbers, they, specific, type, lucas, sequence, displaystyle, which, defined, similar, recurrence, relation, simple, terms, se. In mathematics the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal Like the related Fibonacci numbers they are a specific type of Lucas sequence U n P Q displaystyle U n P Q for which P 1 and Q 2 1 and are defined by a similar recurrence relation in simple terms the sequence starts with 0 and 1 then each following number is found by adding the number before it to twice the number before that The first Jacobsthal numbers are 0 1 1 3 5 11 21 43 85 171 341 683 1365 2731 5461 10923 21845 43691 87381 174763 349525 sequence A001045 in the OEIS A Jacobsthal prime is a Jacobsthal number that is also prime The first Jacobsthal primes are 3 5 11 43 683 2731 43691 174763 2796203 715827883 2932031007403 768614336404564651 201487636602438195784363 845100400152152934331135470251 56713727820156410577229101238628035243 sequence A049883 in the OEIS Contents 1 Jacobsthal numbers 2 Jacobsthal Lucas numbers 3 Jacobsthal Oblong numbers 4 ReferencesJacobsthal numbers editJacobsthal numbers are defined by the recurrence relation J n 0 if n 0 1 if n 1 J n 1 2 J n 2 if n gt 1 displaystyle J n begin cases 0 amp mbox if n 0 1 amp mbox if n 1 J n 1 2J n 2 amp mbox if n gt 1 end cases nbsp The next Jacobsthal number is also given by the recursion formula J n 1 2 J n 1 n displaystyle J n 1 2J n 1 n nbsp or by J n 1 2 n J n displaystyle J n 1 2 n J n nbsp The second recursion formula above is also satisfied by the powers of 2 The Jacobsthal number at a specific point in the sequence may be calculated directly using the closed form equation 2 J n 2 n 1 n 3 displaystyle J n frac 2 n 1 n 3 nbsp The generating function for the Jacobsthal numbers is x 1 x 1 2 x displaystyle frac x 1 x 1 2x nbsp The sum of the reciprocals of the Jacobsthal numbers is approximately 2 7186 slightly larger than e The Jacobsthal numbers can be extended to negative indices using the recurrence relation or the explicit formula giving J n 1 n 1 J n 2 n displaystyle J n 1 n 1 J n 2 n nbsp see OEIS A077925 The following identity holds 2 n J n J n 3 J n 2 displaystyle 2 n J n J n 3J n 2 nbsp see OEIS A139818 Jacobsthal Lucas numbers editJacobsthal Lucas numbers represent the complementary Lucas sequence V n 1 2 displaystyle V n 1 2 nbsp They satisfy the same recurrence relation as Jacobsthal numbers but have different initial values j n 2 if n 0 1 if n 1 j n 1 2 j n 2 if n gt 1 displaystyle j n begin cases 2 amp mbox if n 0 1 amp mbox if n 1 j n 1 2j n 2 amp mbox if n gt 1 end cases nbsp The following Jacobsthal Lucas number also satisfies 2 j n 1 2 j n 3 1 n displaystyle j n 1 2j n 3 1 n nbsp The Jacobsthal Lucas number at a specific point in the sequence may be calculated directly using the closed form equation 2 j n 2 n 1 n displaystyle j n 2 n 1 n nbsp The first Jacobsthal Lucas numbers are 2 1 5 7 17 31 65 127 257 511 1025 2047 4097 8191 16385 32767 65537 131071 262145 524287 1048577 sequence A014551 in the OEIS Jacobsthal Oblong numbers editThe first Jacobsthal Oblong numbers are 0 1 3 15 55 231 903 3655 14535 58311 sequence A084175 in the OEIS J o n J n J n 1 displaystyle Jo n J n J n 1 nbsp References edit Weisstein Eric W Jacobsthal Number MathWorld a b c Sloane N J A ed Sequence A014551 Jacobsthal Lucas numbers The On Line Encyclopedia of Integer Sequences OEIS Foundation Retrieved from https en wikipedia org w index php title Jacobsthal number amp oldid 1181388891 Jacobsthal Lucas numbers, wikipedia, wiki, book, books, library,

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