OFFSET
1,5
COMMENTS
a(n) + A116922(n) = n. For n>= 3, A116922(n) - a(n) is 1 if n is odd, is 2 if n is a multiple of 4 and is 4 if n is congruent to 2 (mod 4).
The arithmetic function v+-(n,2) as defined in A290988. - Robert Price, Aug 22 2017
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
For n >= 3, a(n) = (n-1)/2 if n is odd, a(n) = n/2 - 1 if n is a multiple of 4 and a(n) = n/2 - 2 if n is congruent to 2 (mod 4).
a(n) = (2*n-4-2*(-1)^n+(-1)^(n/2)+(-1)^(3*n/2))/4, n>2. - Wesley Ivan Hurt, May 26 2015
For n > 2, a(n) = (n-2+cos(n*Pi/2)-cos(n*Pi))/2. - Wesley Ivan Hurt, Oct 02 2017
G.f.: t^2*(1+t^3-2*t^4+2*t^5)/((1-t)*(1-t^4)). - Mamuka Jibladze, Aug 22 2019
MATHEMATICA
Join[{0, 1}, Table[(2 n - 4 - 2 (-1)^n + (-1)^(n/2) + (-1)^(3 n/2))/4, {n, 3, 50}]] (* Wesley Ivan Hurt, May 26 2015 *)
Table[Which[OddQ[n], (n-1)/2, Divisible[n, 4], n/2-1, Mod[n, 4]==2, n/2-2], {n, 80}]//Abs (* Harvey P. Dale, Jun 24 2017 *)
PROG
(Magma) [0] cat [(2*n-4-2*(-1)^n+(-1)^(n div 2)+(-1)^(3*n div 2)) div 4: n in [3..80]]; // Vincenzo Librandi, May 26 2015
(PARI) a(n) = {forstep(k = n\2, 0, -1, if (gcd(n, k) == 1, return (k)); ); } \\ Michel Marcus, May 26 2015
(PARI) a(n) = {if(n%2, (n-1)/2, if(n==2, 1, n/2 - if(n%4, 2, 1)))} \\ Andrew Howroyd, Aug 22 2019
(Python)
def A116921(n): return n>>1 if n&1 or n==2 else (n>>1)-(2 if n&2 else 1) # Chai Wah Wu, Jul 31 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Leroy Quet, Feb 26 2006
EXTENSIONS
More terms from Wyatt Lloyd (wal118(AT)psu.edu), Mar 25 2006
STATUS
approved