A007420 Berstel sequence: a(n+1) = 2*a(n) - 4*a(n-1) + 4*a(n-2). (Formerly M0030)

0, 0, 1, 2, 0, -4, 0, 16, 16, -32, -64, 64, 256, 0, -768, -512, 2048, 3072, -4096, -12288, 4096, 40960, 16384, -114688, -131072, 262144, 589824, -393216, -2097152, -262144, 6291456, 5242880, -15728640, -27262976, 29360128, 104857600, -16777216

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

A007420 := proc(n) options remember; if n <=1 then 0 elif n=2 then 1 else 2*A007420(n-1)-4*A007420(n-2)+4*A007420(n-3); fi; end;

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

Cf. A035302, A077953.

Sequence in context: A265829 A022896 A100225 * A362368 A019219 A019139

Adjacent sequences: A007417 A007418 A007419 * A007421 A007422 A007423

Keyword

sign,easy,nice,changed

Author

N. J. A. Sloane

Status

approved