OFFSET
0,4
COMMENTS
a(n) = 0 only for n = 0,1,4,6,13 and 52. [Cassels, following Mignotte. See also Beukers] - N. J. A. Sloane, Aug 29 2010
REFERENCES
J. W. S. Cassels, Local Fields, Cambridge, 1986, see p. 67.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 28.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 193.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..500
F. Beukers, The zero-multiplicity of ternary recurrences, Compositio Math. 77 (1991), 165-177.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Linear recurrence sequences with indices in arithmetic progression and their sums, arXiv:1505.06339 [math.NT], 2015.
M. Mignotte, Suites récurrentes linéaires, Sém. Delange-Pisot-Poitou, 15th year (1973/1974), No. 14, 9 pages.
G. Myerson and A. J. van der Poorten, Some problems concerning recurrence sequences, Amer. Math. Monthly 102 (1995), no. 8, 698-705.
Index entries for linear recurrences with constant coefficients, signature (2,-4,4).
FORMULA
G.f.: x^2/(1-2*x+4*x^2-4*x^3).
a(0)=0, a(1)=0, a(2)=1, a(n) = 2*a(n-1)-4*a(n-2)+4*a(n-3). - Harvey P. Dale, Jun 24 2015
MAPLE
MATHEMATICA
a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = 2*a[n - 1] - 4*a[n - 2] + 4*a[n - 3]; a /@ Range[0, 34] (* Jean-François Alcover, Apr 06 2011 *)
LinearRecurrence[{2, -4, 4}, {0, 0, 1}, 40] (* Harvey P. Dale, Oct 24 2011 *)
Table[RootSum[-4 + 4 # - 2 #^2 + #^3 &, 6 #^n - #^(n + 1) + 4 #^(n + 1) &]/44, {n, 0, 20}] (* Eric W. Weisstein, Nov 09 2017 *)
PROG
(Haskell)
a007420 n = a007420_list !! n
a007420_list = 0 : 0 : 1 : (map (* 2) $ zipWith (+) (drop 2 a007420_list)
(map (* 2) $ zipWith (-) a007420_list (tail a007420_list)))
-- Reinhard Zumkeller, Oct 21 2011
(Magma) I:=[0, 0, 1]; [n le 3 select I[n] else 2*Self(n-1)-4*Self(n-2)+4*Self(n-3): n in [1..70]]; // Vincenzo Librandi, Oct 05 2015
(PARI) a(n)=([0, 1, 0; 0, 0, 1; 4, -4, 2]^n*[0; 0; 1])[1, 1] \\ Charles R Greathouse IV, Feb 19 2017
CROSSREFS
KEYWORD
sign,easy,nice
AUTHOR
STATUS
approved