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A007910 Expansion of 1/((1-2*x)*(1+x^2)). 14
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
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.
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
Sequence in context: A086514 A079662 A290991 * A293315 A052702 A058766
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

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Last modified February 21 04:08 EST 2024. Contains 370219 sequences. (Running on oeis4.)