A093614 Numbers n such that F_n(x) and F_n(1-x) have a common factor mod 2, with F_n(x) = U(n-1,x/2) the monic Chebyshev polynomials of second kind.

5, 6, 10, 12, 15, 17, 18, 20, 24, 25, 30, 31, 33, 34, 35, 36, 40, 42, 45, 48, 50, 51, 54, 55, 60, 62, 63, 65, 66, 68, 70, 72, 75, 78, 80, 84, 85, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 114, 115, 119, 120, 124, 125, 126, 127, 129, 130, 132, 135, 136, 138

Offset

1,1

Comments

Goldwasser et al. proved that 2^k+-1 belongs to the set, for k>4.

Closed under multiplication by positive integers. - Don Knuth, May 11 2006

Links

Ralf Stephan and Thomas Buchholz, Table of n, a(n) for n = 1..1000 [terms 1 through 61 were computed by Ralf Stephan, May 22 2004; terms 62 through 1000 by Thomas Buchholz, May 16 2014]

M. Hunziker, A. Machiavelo and J. Park, Chebyshev polynomials over finite fields and reversibility of sigma-automata on square grids, Theoretical Comp. Sci., 320 (2004), 465-483.

K. Sutner, Linear cellular automata and the Garden-of-Eden, Math. Intelligencer, 11 (No. 2, 1989), 49-53.

K. Sutner, The computational complexity of cellular automata, in Lect. Notes Computer Sci., 380 (1989), 451-459.

Eric Weisstein's World of Mathematics, Lights-Out Puzzle

Prog

(PARI)
{ F2(n)=local(t, t1, t2, tmp); t1=Mod(0, 2); t2=Mod(1, 2); t=Mod(1, 2)*x; for(k=2, n, tmp=t*t2-t1; t1=t2; t2=tmp); tmp }
for(n=2, 200, t=F2(n); if(gcd(t, subst(t, x, 1-x))!=1, print1(n", ")))

Crossrefs

Equals A117870(n) + 1.

Cf. A094425 (primitive elements), A076436.

Sequence in context: A326418 A037359 A099538 * A353280 A093509 A105953

Adjacent sequences: A093611 A093612 A093613 * A093615 A093616 A093617

Keyword

nonn

Author

Ralf Stephan, May 22 2004

Extensions

More terms from Thomas Buchholz, May 16 2014

Status

approved