|
%I M2653 #33 Sep 08 2022 08:44:35
%S 1,3,7,15,31,60,113,207,373,663,1167,2038,3537,6107,10499,17983,30703,
%T 52272,88769,150407,254321,429223,723167,1216490,2043361,3427635,
%U 5742463,9609327,16062463,26821668,44744657,74576703,124192237,206650167,343594479
%N Patterns in a dual ring.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007574/b007574.txt">Table of n, a(n) for n = 1..1000</a>
%H C. A. Church, Jr., <a href="http://www.fq.math.ca/Scanned/12-4/church.pdf">Lattice paths and Fibonacci and Lucas numbers</a>, Fibonacci Quarterly 12(4) (1974) 336-338.
%H W. Dotson, F. Norwood and C. Taylor, <a href="http://www.jstor.org/stable/2690962">Fiber optics and Fibonacci</a>, Math. Mag., 66 (1993), 167-174.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4, -4, -2, 4, 0, -1).
%F a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-3) +4*a(n-4) -a(n-6). G.f.: -x*(-1+x+x^2-x^3-x^4+2*x^5)/ ((x-1)^2 * (x^2+x-1)^2). [_R. J. Mathar_, Feb 06 2010]
%p with(combinat): A007574 := proc(n) local k; if n=1 then RETURN(1) fi; if n=2 then RETURN(3) fi; if n=3 then RETURN(7) fi; if n>3 then RETURN( fibonacci(n)+2*fibonacci(n-1)+n*sum(fibonacci(n-k),k=2..n-1)) fi; end;
%t Table[ Fibonacci[n] + 2 Fibonacci[n - 1] + n*Sum[Fibonacci[n - k], {k, 2, n - 1}], {n, 1, 35} ]
%t LinearRecurrence[{4, -4, -2, 4, 0, -1}, {1, 3, 7, 15, 31, 60}, 60] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2012 *)
%t CoefficientList[Series[-(- 1 + x + x^2 - x^3 - x^4 + 2 x^5) / ((x - 1)^2 (x^2 + x - 1)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%o (Magma) I:=[1,3,7,15,31,60]; [n le 6 select I[n] else 4*Self(n-1)-4*Self(n-2)-2*Self(n-3)+4*Self(n-4)-Self(n-6): n in [1..50]]; // _Vincenzo Librandi_, Jun 09 2013
%Y Cf. A000045.
%K nonn,easy
%O 1,2
%A _Simon Plouffe_ and _N. J. A. Sloane_, _Robert G. Wilson v_
|
