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:
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]
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,