Jacobstal numbers

Jacobstal numbers are an named after the German mathematician E. E. Jacobstal .

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Jacobstal numbers

Like Fibonacci numbers , Jacobstal numbers are one of Luke's sequences

U_{n}(P,Q),

{\ displaystyle U_ {n} (P, Q),}

for which P = 1 and Q = −2 ^[1] . The sequence begins with numbers ^[1] ^[2]

0, 1, 1, 3 , 5 , 11 , 21 , 43 , 85 , 171, 341 , 683, 1365, 2731, 5461, 10 923, 21 845, 43 691, 87 381, 174 763, 349 525, ...

Jacobstal numbers are determined by the recurrence relation ^[1] ^[2]

J_{n}={\begin{cases}0,&n=0;\\1,&n=1;\\J_{n-1}+2J_{n-2},&n>1.\\\end{cases}}

{\ displaystyle J_ {n} = {\ begin {cases} 0, & n = 0; \\ 1, & n = 1; \\ J_ {n-1} + 2J_ {n-2}, & n> 1. \\ \ end {cases}}}

Other options for recursive sequence assignment ^[2] :

$J_{n+1}=2J_{n}+(-1)^{n}$ ${\ displaystyle J_ {n + 1} = 2J_ {n} + (- 1) ^ {n}}$
$J_{n+1}=2^{n}-J_{n}$ ${\ displaystyle J_ {n + 1} = 2 ^ {n} -J_ {n}}$

The Jacobstal number with a given number can be calculated using the formula ^[1] ^[2]

J_{n}={\frac {2^{n}-(-1)^{n}}{3}}.

{\ displaystyle J_ {n} = {\ frac {2 ^ {n} - (- 1) ^ {n}} {3}}.}

Jacobstal-Luc numbers

Jacobstal-Luc numbers represent the sequence of Luc $V_{n}(1,-2)$ ${\ displaystyle V_ {n} (1, -2)}$ . They satisfy the same recurrence relations as the Jacobstal numbers, but differ in initial values ^[1] :

j_{n}={\begin{cases}2,&n=0;\\1,&n=1;\\j_{n-1}+2j_{n-2},&n>1.\\\end{cases}}

{\ displaystyle j_ {n} = {\ begin {cases} 2, & n = 0; \\ 1, & n = 1; \\ j_ {n-1} + 2j_ {n-2}, & n> 1. \\ \ end {cases}}}

Alternative formula ^[3] :

j_{n+1}=2j_{n}-3(-1)^{n}.

{\ displaystyle j_ {n + 1} = 2j_ {n} -3 (-1) ^ {n}.}

The Jacobstal-Luc number with a given number can be calculated using the formula ^[3]

j_{n}=2^{n}+(-1)^{n}.

{\ displaystyle j_ {n} = 2 ^ {n} + (- 1) ^ {n}.}

The Jacobstal-Luc sequence begins with the numbers ^[1] ^[3]

2, 1, 5 , 7 , 17 , 31 , 65 , 127 , 257 , 511 , 1025, 2047, 4097, 8191, 16 385, 32 767, 65 537 , 131 071, 262 145, 524 287, 1,048 577, ...

Notes

↑ ¹ ² ³ ⁴ ⁵ ⁶ Weisstein, Eric W. Jacobsthal Number on the Wolfram MathWorld website.
↑ ¹ ² ³ ⁴ A001045 sequence in OEIS = Jacobsthal sequence
↑ ¹ ² ³ Sequence A014551 in OEIS = Jacobsthal-Lucas numbers

Literature

AF Horadam . Jacobsthal representation numbers (English) (May 1994).
Paul Barry . Triangle Geometry and Jacobsthal Numbers (English) , Irish Math. Soc. Bulletin (April 2003).
Zvonko Čerin . Sums of Squares and Products of Jacobsthal Numbers , Journal of Integer Sequences.

Links

Sequence

A049883 at OEIS : Jacobstal Primes

[mw-1] ¹ ² ³ ⁴ ⁵ ⁶ Weisstein, Eric W. Jacobsthal Number on the Wolfram MathWorld website.

[oeis-a001045-2] ¹ ² ³ ⁴ A001045 sequence in OEIS = Jacobsthal sequence

[oeis-a014551-3] ¹ ² ³ Sequence A014551 in OEIS = Jacobsthal-Lucas numbers

Jacobstal numbers

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