OFFSET
0,4
COMMENTS
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = -1, y = -2, z = -3.
The Hankel transform of -A026726(n+k) is a(2*n+2+k) for k = 0, 1.
Let a(n) := F(n) * (-1)^binomial(n, 2). Then a(m - n) * a(m + n) = a(m + 1) * a(m - 1) * a(n)^2 - a(n + 1) * a(n - 1) * a(m)^2. This plus gcd(f[n], f[m]) = |f[gcd(n, m)]| makes a[] a strong elliptic divisibility sequence. Likewise F(n) * (-1)^binomial(n - 1, 2), but no other asSIGNation (mod scaling). - Bill Gosper, May 28 2008
LINKS
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
Index entries for linear recurrences with constant coefficients, signature (0,-3,0,-1).
FORMULA
G.f.: (x^3 - x^2 + x)/(x^4 + 3*x^2 + 1).
a(n) = -a(-n) = -3*a(n+2) -a(n+4) for all n in Z.
0 = a(n)^2 -a(n+1)^2 +a(n+2)^2 +2*a(n)*a(n+2) for all n in Z.
0 = a(n)*(+a(n+2)) +a(n+1)*(+a(n+1) +a(n+3)) +a(n+2)*(+a(n+2)) for all n in Z.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) - 2*a(n+2)^2 for all n in Z.
0 = a(n)*a(n+5) + 2*a(n+1)*a(n+4) - 3*a(n+2)*a(n+3) for all n in Z.
a(n+1) = i^(n^2) * U(n, i/2) for all n in Z [From Gosper, Mar 19 2020]. - Michael Somos, Mar 19 2020
EXAMPLE
G.f. = x - x^2 - 2*x^3 + 3*x^4 + 5*x^5 - 8*x^6 - 13*x^7 + 21*x^8 + ...
MATHEMATICA
a[ n_] := Fibonacci[n] (-1)^(n (n - 1) / 2);
a[ n_] := With[{m=n-1}, I^m^2 ChebyshevU[m, I/2]]; (* Michael Somos, Mar 19 2020 *)
PROG
(PARI) {a(n) = fibonacci(n) * (-1)^(n*(n-1)/2)};
(Sage)
def A333378():
a, b, c, d = False, True, True, False
x, y = 0, 1
while True:
yield x if a else -x
x, y = y, x - y
a, b, c, d = b, c, d, a
a = A333378()
print([next(a) for _ in range(39)]) # Peter Luschny, Mar 19 2020
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Michael Somos, Mar 17 2020
STATUS
approved