login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

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

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: A113050 A278180 A015927 * A116961 A120611 A092063

Adjacent sequences:  A097107 A097108 A097109 * A097111 A097112 A097113

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jul 25 2004, corrected Sep 05 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 4 09:05 EST 2016. Contains 278749 sequences.