A007679 If n mod 4 = 0 then 2^(n-1)+1 elif n mod 4 = 2 then 2^(n-1)-1 else 2^(n-1). (Formerly M3359)

1, 1, 4, 9, 16, 31, 64, 129, 256, 511, 1024, 2049, 4096, 8191, 16384, 32769, 65536, 131071, 262144, 524289, 1048576, 2097151, 4194304, 8388609, 16777216, 33554431, 67108864, 134217729, 268435456, 536870911

Offset

1,3

References

M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 37. [From N. J. A. Sloane, Jan 29 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

I. Nemes et al., How to do Monthly problems with your computer, Amer. Math. Monthly, 104 (1997), 505-519.

Index entries for linear recurrences with constant coefficients, signature (2,-1,2).

Formula

a(n) = 2^(n-1) + cos(n*Pi/2).

a(n) = sum(2^k*C(n-k, 2k)*n/(n-k), k=0..floor(n/3)).

a(n) = A007909(n) + A007910(n).

a(n) = ((-i)^n+i^n+2^n)/2, where i=sqrt(-1). a(n) = 2*a(n-1)-a(n-2)+2*a(n-3). G.f.: x*(1-x+3*x^2)/((1-2*x)*(1+x^2)). [Colin Barker, May 08 2012]

Maple

f:=n->2^(n-1)+cos(Pi*n/2);

Mathematica

CoefficientList[Series[(1-x+3*x^2)/((1-2*x)*(1+x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, May 09 2012 *)
Table[Which[Mod[n, 4]==0, 2^(n-1)+1, Mod[n, 4]==2, 2^(n-1)-1, True, 2^(n-1)], {n, 30}] (* or *) LinearRecurrence[{2, -1, 2}, {1, 1, 4}, 30] (* Harvey P. Dale, May 01 2018 *)

Prog

(Magma) I:=[1, 1, 4]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, May 09 2012

Crossrefs

Sequence in context: A199936 A326958 A281904 * A239870 A068037 A167188

Adjacent sequences: A007676 A007677 A007678 * A007680 A007681 A007682

Keyword

nonn,easy

Author

N. J. A. Sloane, R. K. Guy, Simon Plouffe.

Status

approved