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

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A007909, A007679.

Programs

Magma

[Round(2^(n+2)/5): n in [0..40]]; // Vincenzo Librandi, Jun 21 2011

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 *)

PARI

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

Python

def A007910(n): return ((4<Chai Wah Wu, Apr 17 2025

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

Extensions

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

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