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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

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

Links

• 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).

Crossrefs

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.

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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)

Python

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

SageMath

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

Formula

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

Extensions

More terms from Michael Somos, Aug 07 2000