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A007910 Expansion of 1/((1-2*x)*(1+x^2)). 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

a(n) = 2a(n-1)-a(n-2)+2a(n-3). Sequence is identical to its half second differences from the second term; a(n)+a(n+2)=2^(n+2). - Paul Curtz, Dec 17 2007

REFERENCES

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

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

LINKS

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

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

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(1) = 1, a(2n+1) = 2*a(2n) and a(2n) = 2*a(2n-1) + (-1)^n.

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

a(n) = round(2^(n+1)/5). [Mircea Merca, Dec 27 2010]

MAPLE

V:=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+1)/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+1)/5): n in [1..40]]; // Vincenzo Librandi, Jun 21 2011

(PARI) a(n)=2^(n+1)\/5 \\ Charles R Greathouse IV, Jun 21 2011

CROSSREFS

Cf. A007909, A007679.

Sequence in context: A018775 A086514 A079662 * A052702 A058766 A127601

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

STATUS

approved

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Last modified May 24 22:11 EDT 2017. Contains 287008 sequences.