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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 The same sequence is obtained iteratively by setting a(0)=1, a(1)=2, and a(n) = a(n-1) + a(n-2) - (GCD(1+a(n-1), a(n-2))-1), showing a kind of affinity to Fibonacci numbers. - Stanislav Sykora, Oct 16 2015 LINKS Index entries for 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 MAPLE seq(op([2^n-1, 2^n]), n=1..100); # Robert Israel, Oct 16 2015 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 *) PROG (PARI) a=vector(1000); a[0]=1; a[1]=2; for(n=3, #a, a[n]=a[n-1]+a[n-2]+1-gcd(1+a[n-1], a[n-2])) \\ Stanislav Sykora, Oct 16 2015 (PARI) Vec(2*(1+x)/(1-2*x^2)-1/(1-x^2) + O(x^100)) \\ Altug Alkan, Oct 16 2015 CROSSREFS Cf. A000045, A000079, A000225. 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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