A007910 Expansion of 1/((1-2*x)*(1+x^2)).

1, 2, 3, 6, 13, 26, 51, 102, 205, 410, 819, 1638, 3277, 6554, 13107, 26214, 52429, 104858, 209715, 419430, 838861, 1677722, 3355443, 6710886, 13421773, 26843546, 53687091, 107374182, 214748365, 429496730, 858993459, 1717986918, 3435973837, 6871947674

Offset

0,2

Comments

Also describes the location a(n) of the minimal scaling factor when rescaling an FFT of order 2^{n+2} in order to (currently) minimize the arithmetic operation count (Johnson & Frigo, 2007). - Steven G. Johnson (stevenj(AT)math.mit.edu), Dec 27 2006

References

M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 38.

Links

M. F. Hasler, Table of n, a(n) for n = 0..1000 (in replacement of a(0..999) indexed 1..1000 by Vincenzo Librandi)

M. H. Cilasun, An Analytical Approach to Exponent-Restricted Multiple Counting Sequences, arXiv preprint arXiv:1412.3265 [math.NT], 2014.

M. H. Cilasun, Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.

I. Gessel, Problem 10424, Amer. Math. Monthly, 102 (1995), 70.

S. G. Johnson and M. Frigo, A modified split-radix FFT with fewer arithmetic operations, IEEE Trans. Signal Processing 55 (2007), 111-119.

Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

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

Formula

a(0) = 1, a(2n+1) = 2*a(2n) and a(2n) = 2*a(2n-1) + (-1)^n. [Corrected by M. F. Hasler, Feb 22 2018]

a(n) = (4*2^n+cos(Pi*n/2)+2*sin(Pi*n/2))/5. - Paul Barry, Dec 17 2003

a(n) = 2a(n-1)-a(n-2)+2a(n-3). Sequence equals half its second differences with first term dropped. a(n) + a(n+2) = 2^(n+2). - Paul Curtz, Dec 17 2007

a(n) = round(2^(n+2)/5). - Mircea Merca, Dec 27 2010

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k). - Gerry Martens, Oct 15 2022

Maple

A007910:=n->(1/5)*(2^(n-1)+2*cos(n*Pi/2)-sin(n*Pi/2)); [seq(V(n), n=0..12)];
seq(round(2^(n+2)/5), n=1..25) # Mircea Merca, Dec 27 2010

Mathematica

CoefficientList[Series[1/((1 - 2 x) (1 + x^2)), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
LinearRecurrence[{2, -1, 2}, {1, 2, 3}, 40] (* Harvey P. Dale, Feb 22 2016 *)

Prog

(Magma) [Round(2^(n+2)/5): n in [0..40]]; // Vincenzo Librandi, Jun 21 2011
(PARI) a(n)=2^(n+2)\/5 \\ Charles R Greathouse IV, Jun 21 2011

Crossrefs

Cf. A007909, A007679.

Sequence in context: A086514 A079662 A290991 * A293315 A052702 A058766

Adjacent sequences: A007907 A007908 A007909 * A007911 A007912 A007913

Keyword

nonn,easy

Author

Mogens Esrom Larsen (mel(AT)math.ku.dk)

Extensions

Entry revised by N. J. A. Sloane, Feb 24 2004

Offset corrected and minor edits by M. F. Hasler, Feb 22 2018

Status

approved