A333378 - OEIS

%I #32 Jan 05 2025 19:51:41

%S 0,1,-1,-2,3,5,-8,-13,21,34,-55,-89,144,233,-377,-610,987,1597,-2584,

%T -4181,6765,10946,-17711,-28657,46368,75025,-121393,-196418,317811,

%U 514229,-832040,-1346269,2178309,3524578,-5702887,-9227465,14930352,24157817,-39088169

%N a(n) = F(n) * (-1)^(n*(n-1)/2) where F(n) = A000045(n) Fibonacci numbers.

%C This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = -1, y = -2, z = -3.

%C The Hankel transform of -A026726(n+k) is a(2*n+2+k) for k = 0, 1.

%C 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

%H Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,-3,0,-1).

%F G.f.: (x^3 - x^2 + x)/(x^4 + 3*x^2 + 1).

%F a(n) = -a(-n) = -3*a(n+2) -a(n+4) for all n in Z.

%F 0 = a(n)^2 -a(n+1)^2 +a(n+2)^2 +2*a(n)*a(n+2) for all n in Z.

%F 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.

%F 0 = a(n)*a(n+4) - a(n+1)*a(n+3) - 2*a(n+2)^2 for all n in Z.

%F 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.

%F 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

%e G.f. = x - x^2 - 2*x^3 + 3*x^4 + 5*x^5 - 8*x^6 - 13*x^7 + 21*x^8 + ...

%t a[ n_] := Fibonacci[n] (-1)^(n (n - 1) / 2);

%t a[ n_] := With[{m=n-1}, I^m^2 ChebyshevU[m, I/2]]; (* _Michael Somos_, Mar 19 2020 *)

%o (PARI) {a(n) = fibonacci(n) * (-1)^(n*(n-1)/2)};

%o (Sage)

%o def A333378():

%o     a, b, c, d = False, True, True, False

%o     x, y = 0, 1

%o     while True:

%o         yield x if a else -x

%o         x, y = y, x - y

%o         a, b, c, d = b, c, d, a

%o a = A333378()

%o print([next(a) for _ in range(39)]) # _Peter Luschny_, Mar 19 2020

%Y Cf. A000045, A026726.

%K sign,easy

%O 0,4

%A _Michael Somos_, Mar 17 2020