A066043 -id:A066043 - cp's OEIS frontend

A022998 If n is odd then n, otherwise 2n.

0, 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128, 65, 132, 67

Offset: 0

N. J. A. Sloane

Also for n > 0: numerator of Sum_{i=1..n} 2/(i*(i+1)), denominator=A026741. - Reinhard Zumkeller, Jul 25 2002
For n > 2: a(n) = gcd(A143051((n-1)^2), A143051(1+(n-1)^2)) = A050873(A000290(n-1), A002522(n-1)). - Reinhard Zumkeller, Jul 20 2008
Partial sums give the generalized octagonal numbers A001082. - Omar E. Pol, Sep 10 2011
Multiples of 4 and odd numbers interleaved. - Omar E. Pol, Sep 25 2011
The Pisano period lengths modulo m appear to be A066043(m). - R. J. Mathar, Oct 08 2011
The partial sums a(n)/A026741(n+1) given by R. Zumkeller in a comment above are 2*n/(n+1) (telescopic sum), and thus converge to 2. - Wolfdieter Lang, Apr 09 2013
a(n) = numerator(H(n,1)), where H(n,1) = 2*n/(n+1) is the harmonic mean of 1 and n. a(n) = 2*n/gcd(2n, n+1) = 2*n/gcd(n+1,2). a(n) = A227041(n,1), n>=1. - Wolfdieter Lang, Jul 04 2013
a(n) = numerator of the mean (2n/(n+1), after reduction), of the compositions of n; denominator is given by A001792(n-1). - Clark Kimberling, Mar 11 2014
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of numerators. The sequence of denominators of the continued fraction convergents [1, 1, 3, 2, 5, 3, 7, ...] is A026741, also a strong divisibility sequence. Cf. A203976. - Peter Bala, May 19 2014
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized octagonal numbers. - Omar E. Pol, Jul 27 2018
a(n) is the number of petals of the Rhodonea curve r = a*cos(n*theta) or r = a*sin(n*theta). - Matt Westwood, Nov 19 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rose
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A026741, A059026, A203976, A345082.
Column 4 of A195151. - Omar E. Pol, Sep 25 2011
Cf. A000034, A001082 (partial sums).
Cf. A227041 (first column). - Wolfdieter Lang, Jul 04 2013
Row 2 of A349593. A385555, A385556, A385557, A385558, A385559, and A385560 are respectively rows 3, 4, 5-6, 7, 8, and 9-10.

a022998 n = a000034 (n + 1) * n
a022998_list = zipWith (*) [0..] $ tail a000034_list
-- Reinhard Zumkeller, Mar 31 2012

[((-1)^n+3)*n/2: n in [0..70]]; // Vincenzo Librandi, Sep 17 2011

A022998 := proc(n) if type(n,'odd') then n ; else 2*n; end if; end proc: # R. J. Mathar, Mar 10 2011

Table[n (3 + (-1)^n)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 13 2013 *)
Table[If[OddQ[n],n,2n],{n,0,150}] (* or *) Riffle[ 2*Range[ 0,150,2], Range[ 1,150,2]] (* Harvey P. Dale, Feb 06 2017 *)

PARI
```
a(n)=if(n%2,n,2*n)
```

def A022998(n): return n if n&1 else n<<1 # Chai Wah Wu, Mar 05 2024

[n*(1+((n+1)%2)) for n in (0..80)] # G. C. Greubel, Jul 31 2022

Denominator of (n+1)*(n-1)*(2*n+1)/(2*n) (for n > 0).
a(n+1) = lcm(n, n+2)/n + lcm(n, n+2)/(n+2) for all n >= 1. - Asher Auel, Dec 15 2000
Multiplicative with a(2^e) = 2^(e+1), a(p^e) = p^e, p > 2.
G.f. x*(1 + 4*x + x^2)/(1-x^2)^2. - Ralf Stephan, Jun 10 2003
a(n) = 3*n/2 + n*(-1)^n/2 = n*(3 + (-1)^n)/2. - Paul Barry, Sep 04 2003
a(n) = A059029(n-1) + 1 = A043547(n+2) - 2.
a(n)*a(n+3) = -4 + a(n+1)*a(n+2).
a(n) = n*(((n+1) mod 2) + 1) = n^2 + 2*n - 2*n*floor((n+1)/2). - William A. Tedeschi, Feb 29 2008
a(n) = denominator((n+1)/(2*n)) for n >= 1; A026741(n+1) = numerator((n+1)/(2*n)) for n >= 1. - Johannes W. Meijer, Jun 18 2009
a(n) = 2*a(n-2) - a(n-4).
Dirichlet g.f. zeta(s-1)*(1+2^(1-s)). - R. J. Mathar, Mar 10 2011
a(n) = n * (2 - n mod 2) = n * A000034(n+1). - Reinhard Zumkeller, Mar 31 2012
a(n) = floor(2*n/(1 + (n mod 2))). - Wesley Ivan Hurt, Dec 13 2013
From Ilya Gutkovskiy, Mar 16 2017: (Start)
E.g.f.: x*(2*sinh(x) + cosh(x)).
It appears that a(n) is the period of the sequence k*(k + 1)/2 mod n. (End) [This is correct; see A349593. - Jianing Song, Jul 03 2025]
a(n) = Sum_{d | n} A345082(d). - Peter Bala, Jan 13 2024

More terms from Michael Somos, Aug 07 2000

A038608 a(n) = n*(-1)^n.

Original entry on oeis.org

0, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59, 60, -61, 62, -63, 64, -65

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998

a(n) is the determinant of the (n+1) X (n+1) matrix with 0's in the main diagonal and 1's elsewhere. - Franz Vrabec, Dec 01 2007
Sum_{n>0} 1/a(n) = -log(2). - Jaume Oliver Lafont, Feb 24 2009
Pisano period lengths:  1, 2, 6, 4, 10, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 18, 38, 20, ... (is this A066043?). - R. J. Mathar, Aug 10 2012
a(n) is the determinant of the (n+1) X (n+1) matrix whose i-th row, j-th column entry is the value of the cubic residue symbol ((j-i)/p) where p is a prime of the form 3k+2 and n < p. - Ryan Wood, Nov 09 2017
a(n-1) is the difference in the number of even minus odd parity derangements (permutations with no fixed points) in symmetric group S_n. - Julian Hatfield Iacoponi, Aug 01 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Tanya Khovanova, Recursive Sequences
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A002162 (log(2)).
Cf. A001477.
Cf. A003221, A000387 (even, odd derangements).

a038608 n = n * (-1) ^ n
a038608_list = [0, -1] ++ map negate
   (zipWith (+) a038608_list (map (* 2) $ tail a038608_list))
-- Reinhard Zumkeller, Nov 24 2012

[n*(-1)^n: n in [0..80]]; // Vincenzo Librandi, Jun 08 2011

A038608 := n->n*(-1)^n; seq(A038608(n), n=0..100);

Array[# (-1)^# &, 66, 0] (* Michael De Vlieger, Nov 18 2017 *)
Table[If[EvenQ[n],n,-n],{n,0,70}] (* Harvey P. Dale, Jan 17 2022 *)

a(n)=n*(-1)^n \\ Charles R Greathouse IV, Dec 07 2011

def A038608(n): return -n if n&1 else n # Chai Wah Wu, Nov 14 2022

G.f.: -x/(1+x)^2.
E.g.f: -x*exp(-x).
a(n) = -2*a(n-1) - a(n-2) for n >= 2. - Jaume Oliver Lafont, Feb 24 2009
a(n) = A003221(n+1)-A000387(n+1). - Julian Hatfield Iacoponi, Aug 01 2024

Edited by Frank Ellermann, Jan 28 2002

A109043 a(n) = lcm(n,2).

Original entry on oeis.org

0, 2, 2, 6, 4, 10, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 18, 38, 20, 42, 22, 46, 24, 50, 26, 54, 28, 58, 30, 62, 32, 66, 34, 70, 36, 74, 38, 78, 40, 82, 42, 86, 44, 90, 46, 94, 48, 98, 50, 102, 52, 106, 54, 110, 56, 114, 58, 118, 60, 122, 62, 126, 64, 130, 66, 134

Offset: 0

Mitch Harris, Jun 18 2005

Exponent of the dihedral group D(2n) = . - Arkadiusz Wesolowski, Sep 10 2013
Second column of table A210530. - Boris Putievskiy, Jan 29 2013
For n > 1, the basic period of A000166(k) (mod n) (Miska, 2016). - Amiram Eldar, Mar 03 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, The Exponent of a Group: Properties, Computations and Applications, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
Piotr Miska, Arithmetic properties of the sequence of derangements, Journal of Number Theory, Vol. 163 (2016), pp. 114-145; arXiv preprint, arXiv:1508.01987 [math.NT], 2015. See p. 124 (p. 14 in the preprint).
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
Index entries for sequences related to lcm's.

Cf. A000166, A109042, A152749 (partial sums).
Cf. A066043 (essentially the same), A000034 (=a(n)/n), A026741 (=a(n)/2).
Cf. A186421, A210530.

a109043 = (lcm 2)
a109043_list = zipWith (*) [0..] a000034_list
-- Reinhard Zumkeller, Mar 31 2012

[0, 2, 2] cat [Exponent(DihedralGroup(n)) : n in [3..65]]; // Arkadiusz Wesolowski, Sep 10 2013

LCM[Range[0,70],2] (* Harvey P. Dale, Aug 19 2012 *)

a(n)=lcm(n,2) \\ Charles R Greathouse IV, Sep 24 2015

def A109043(n): return n<<1 if n&1 else n # Chai Wah Wu, Aug 05 2024

[lcm(n,2) for n in range(0, 68)] # Zerinvary Lajos, Jun 07 2009

For n > 0: a(n) = A186421(n) + A186421(n+2).
a(n) = n*2 / gcd(n, 2).
a(n) = -(n*((-1)^n-3))/2. - Stephen Crowley, Feb 11 2007
From R. J. Mathar, Aug 20 2008: (Start)
a(n) = A066043(n), n > 1.
a(n) = 2*A026741(n).
G.f.: 2*x(1+x+x^2)/((1-x)^2*(1+x)^2). (End)
a(n) = n*A000034(n). - Paul Curtz, Mar 25 2011
E.g.f.: x*(2*cosh(x) + sinh(x)). - Stefano Spezia, May 09 2021
Sum_{k=1..n} a(k) ~ (3/4) * n^2. - Amiram Eldar, Nov 26 2022

A204904 p(n)-q(n), where (p(n), q(n)) is the least pair of odd primes for which n divides p(n)-q(n).

Original entry on oeis.org

2, 2, 6, 4, 10, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 18, 38, 20, 42, 22, 46, 24, 50, 26, 54, 28, 58, 30, 62, 32, 66, 34, 70, 36, 74, 38, 78, 40, 82, 42, 86, 44, 90, 46, 94, 48, 98, 50, 102, 52, 106, 54, 110, 56, 114, 58, 118, 60, 122, 62, 126, 64, 130

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

This sequence agrees with A109043 for 0

For a guide to related sequences, see A204892.

Sequence agrees with A109043 at least up to 6400. - Michel Marcus, Mar 14 2018

If Polignac's conjecture is true, then this is a duplicate of A109043. - Robert Israel, Mar 14 2018

			1 = (5-3)/2=(7-3)/4=(13-3)/6=(11-3)/8=...
2 = (5-3)/1=(11-5)/3=(7-3)/5=(17-3)/7=...

Wikipedia, Polignac's conjecture

Cf. A066043, A109043, A204900, A204892, A210530.

Mathematica
```
(See the program at A204900.)
```

a(n) = {forprime(p=5,,forprime(q=3, p-1, d = p-q; if ((d % n) == 0, return (d));););} \\ Michel Marcus, Mar 14 2018

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A022998 If n is odd then n, otherwise 2n.

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A038608 a(n) = n*(-1)^n.

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A109043 a(n) = lcm(n,2).

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A204904 p(n)-q(n), where (p(n), q(n)) is the least pair of odd primes for which n divides p(n)-q(n).

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