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A097110 Expansion of (1 + 2x - 2x^3) / (1 - 3x^2 + 2x^4). 2
1, 2, 3, 4, 7, 8, 15, 16, 31, 32, 63, 64, 127, 128, 255, 256, 511, 512, 1023, 1024, 2047, 2048, 4095, 4096, 8191, 8192, 16383, 16384, 32767, 32768, 65535, 65536, 131071, 131072, 262143, 262144, 524287, 524288, 1048575, 1048576, 2097151, 2097152 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Union of A000079 and A000225 without 0 = 2^0 - 1. - Reinhard Zumkeller, Jan 18 2005

Let f(0)=1, f(1)=1, and f(n) = f(n - 1 - (1 + (-1)^n)/2) + f(n-2); then a(n-1) = f(n). - John M. Campbell, May 22 2011

LINKS

Table of n, a(n) for n=0..41.

Index to sequences with linear recurrences with constant coefficients, signature (0,3,0,-2).

FORMULA

G.f.: 2*(1+x)/(1-2*x^2)-1/(1-x^2);

a(n) = 3*a(n-2)-2*a(n-4);

a(n) = (1+sqrt(2)/2)*(sqrt(2))^n+(1/2-sqrt(2)/2)*(-sqrt(2))^n-(1+(-1)^n)/2;

a(n) = sum(k=0..n, binomial(floor(n/2), floor(k/2)) ).

a(n) = 2^floor((n+2)/2) - 1 + n mod 2. - Reinhard Zumkeller, Jan 18 2005

MATHEMATICA

t={1}; Do[AppendTo[t, t[[-1]]+1]; AppendTo[t, t[[-1]]+t[[-2]]], {n, 10}]; t (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)

CoefficientList[Series[(1 + 2*x - 2*x^3)/(1 - 3*x^2 + 2*x^4), {x, 0, 40}], x] (* T. D. Noe, Jan 27 2012 *)

CROSSREFS

Sequence in context: A240690 A113050 A015927 * A116961 A120611 A092063

Adjacent sequences:  A097107 A097108 A097109 * A097111 A097112 A097113

KEYWORD

easy,nonn,changed

AUTHOR

Paul Barry, Jul 25 2004, corrected Sep 05 2006

STATUS

approved

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Last modified March 2 02:16 EST 2015. Contains 255130 sequences.