OFFSET
0,3
REFERENCES
Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).
FORMULA
There's an obvious formula for the n-th term!
G.f.: x*(1+x)^2/(1+x^2)^2.
abs(a(n)) = Sum_{k=0..floor((n-1)/2)} (C(n-k-1, k) mod 2)*(-1)^k*2^A000120(n-2k-1). - Paul Barry, Jan 06 2005
a(n) = (n^(n+1) mod (n+1)) * (-1)^[(n-1)/2] = a(n-1)-a(n-2)+(-1)^n*a(n-1) = -2a(n-2)-a(n-4). - Henry Bottomley, May 07 2005
a(n+2) is the Hankel transform of A086622(n+1). - Paul Barry, Nov 06 2007
E.g.f.: sin(x)*(1+x)=x*Q(0); Q(k)=1+x/(1-x/(x-2*(k+1)*(2k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011
a(n) = sin(Pi*n/2)-n*cos(Pi*n/2). - Vaclav Kotesovec, Oct 03 2014
a(n) = (((2*n+3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4)+((2*n-1-(-1)^n)/2)*(-1)^((6*n+5-(-1)^n)/4))/2. - Luce ETIENNE, Jul 18 2015
MATHEMATICA
CoefficientList[Series[x*(1+x)^2/(1+x^2)^2, {x, 0, 100}], x] (* Vaclav Kotesovec, Oct 03 2014 *)
PROG
(PARI) concat(0, Vec(x*(1+x)^2/(1+x^2)^2 + O(x^80))) \\ Michel Marcus, Oct 03 2014
(PARI) A009531(n) = (((n^(n+1)) % (n+1)) * ((-1)^((n-1)\2))); \\ Antti Karttunen, Nov 02 2017, after Henry Bottomley's formula.
(PARI) A009531(n) = (lift(Mod(n, n+1)^(n+1)) * ((-1)^((n-1)\2))); \\ (like above, but quicker) - Antti Karttunen, Nov 02 2017
(Magma) [(((2*n+3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n) div 4)+((2*n-1-(-1)^n) div 2)*(-1)^((6*n+5-(-1)^n) div 4))/2: n in [0..90]]; // Vincenzo Librandi, Jul 19 2015
CROSSREFS
KEYWORD
sign,easy
AUTHOR
STATUS
approved