|
%I M0029 #23 Sep 08 2022 08:44:35
%S 2,0,3,2,6,7,14,20,35,54,90,143,234,376,611,986,1598,2583,4182,6764,
%T 10947,17710,28658,46367,75026,121392,196419,317810,514230,832039,
%U 1346270,2178308,3524579,5702886,9227466,14930351,24157818,39088168
%N Fibonacci(n) - (-1)^n.
%C Graham shows that this sequence is (eventually) complete, that is, any large enough number can be written as a sum of finitely many terms of this sequence, and that it retains this property if any finite number of terms are removed, but loses this property if any infinite number of terms are removed. Contrast with the Fibonacci numbers, which retain the property with loss of any one but lose it with the removal of any two. - _Charles R Greathouse IV_, Dec 20 2013
%D R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 129.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. L. Graham, <a href="http://www.fq.math.ca/Scanned/2-1/graham.pdf">A property of Fibonacci numbers</a>, Fibonacci Quarterly 2:1 (1964), pp. 1-10.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1).
%F G.f.: x*(2-x^2)/((1+x)*(1-x-x^2)).
%F a(n) = 2*(n-2)+a(n-3).
%p with(combinat): A007492 := n->fibonacci(n)-(-1)^n;
%t Table[Fibonacci[n] - (-1)^n, {n, 40}] (* _Bruno Berselli_, Dec 20 2013 *)
%o (PARI) a(n)=fibonacci(n)-(-1)^n
%o (Magma) [(Fibonacci(n)-(-1)^n): n in [1..55]]; // _Vincenzo Librandi_, Apr 23 2011
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E More terms from _Michael Somos_, Apr 28, 2000.
|
