OFFSET
0,3
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 entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
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)*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)
a(n) = Sum_{d | n} A345082(d). - Peter Bala, Jan 13 2024
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 *)
PROG
(PARI) a(n)=if(n%2, n, 2*n)
(Magma) [((-1)^n+3)*n/2: n in [0..70]]; // Vincenzo Librandi, Sep 17 2011
(Haskell)
a022998 n = a000034 (n + 1) * n
a022998_list = zipWith (*) [0..] $ tail a000034_list
-- Reinhard Zumkeller, Mar 31 2012
(SageMath) [n*(1+((n+1)%2)) for n in (0..80)] # G. C. Greubel, Jul 31 2022
(Python)
def A022998(n): return n if n&1 else n<<1 # Chai Wah Wu, Mar 05 2024
CROSSREFS
Column 4 of A195151. - Omar E. Pol, Sep 25 2011
Cf. A227041 (first column). - Wolfdieter Lang, Jul 04 2013
KEYWORD
nonn,easy,mult
AUTHOR
EXTENSIONS
More terms from Michael Somos, Aug 07 2000
STATUS
approved