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

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

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

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).

Crossrefs

Cf. A035302, A077953.

Programs

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

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 *)

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

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