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%I M3258 #60 Sep 08 2022 08:44:34
%S 1,0,4,5,15,28,60,117,230,440,834,1560,2891,5310,9680,17527,31545,
%T 56468,100590,178395,315106,554530,972564,1700400,2964325,5153868,
%U 8938300,15465497,26700915,46004620,79112304,135801105,232715006,398151740
%N Generalized Lucas numbers.
%C For n>2 note that (n+1)|a(n) unless n is prime, in which case (n+1)|2*a(n). This sequence is not the better-known generalized Lucas numbers V(n,a,b) defined for fixed integers a and b such that D = a^2 + 4*b is nonnegative, V(0) = 2, V(1) = a and for n>1 the recurrence V(n) = V(n-1) + V(n-2). The a = b = 1 case gives the Lucas Numbers. - _Jonathan Vos Post_, Mar 16 2005
%C Number of circular binary words of length n+1 having exactly two occurrences of 00. Example: a(4) = 5 because we have 00011, 10001, 11000, 00110 and 01100. Column 2 of A119458. - _Emeric Deutsch_, May 20 2006
%C From _Petros Hadjicostas_, Jan 10 2019: (Start)
%C In view of the comment by _Emeric Deutsch_ above, we clarify the previous comment by _Jonathan Vos Post_. We have that 25 + 1 = 26 does not divide a(25) = 2964325 even though n = 25 is not a prime. What he probably meant is that, for n >= 1, we have (n+1) | a(n) unless n is odd, in which case (n+1)|2*a(n). (Of course, for some odd numbers n, we do have (n+1)|a(n), but not for all of them.)
%C From _Emeric Deutsch_'s comment above, we have a(n) = A119458(n+1, k=2), while from the theory of marked and unmarked circular words, we have A320341(n+1, k=2) = (1/(n+1))*Sum_{d|gcd(n+1, 2)} phi(d)*A119458((n+1)/d, 2/d).
%C If n is even, then n+1 is odd, and hence A320341(n+1, k=2) = (1/(n+1)) * A119458(n+1, 2) = a(n)/(n+1), i.e., (n+1)|a(n).
%C If n=1, then (1+1)|2*a(1). Let n be odd >= 3, in which case n+1 is even and 2*A320341(n+1, k=2) = (1/(n+1))*(2*A119458(n+1, k=2) + 2*A119458((n+1)/2, k=1)). Thus, 2*A320341(n+1, k=2) = (1/(n+1))*(2*a(n) + 2*A006490((n+1)/2)) = (1/(n+1))*(2*a(n) + 2*((n+1)/2)*Fibonacci((n-3)/2)). It follows that 2*A320341(n+1, k=2) = 2*a(n)/(n+1) + Fibonacci((n-3)/2). Thus, 2*a(n)/(n+1) = 2*A320341(n+1, k=2) - Fibonacci((n-3)/2), and thus, (n+1)|2*a(n). (End)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A006491/b006491.txt">Table of n, a(n) for n = 1..1000</a>
%H L. Carlitz and R. Scoville, <a href="http://www.fq.math.ca/Scanned/15-3/carlitz1.pdf">Zero-one sequences and Fibonacci numbers</a>, Fibonacci Quarterly, 15 (1977), 246-254.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-5,0,3,1).
%F G.f.: x*(1-x)*(1-2*x+2*x^2)/(1-x-x^2)^3. - _Ralf Stephan_, Apr 23 2004, corrected Feb 08 2006
%F a(n) = a(n-1) + a(n-2) + n*Fibonacci(n-2) - (n-1)*Fibonacci(n-3) for n >= 3; a(1)=1, a(2)=0. - _Emeric Deutsch_, May 20 2006
%F a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6). - _G. C. Greubel_, Jan 01 2018
%p G:=x*(1-x)*(1-2*x+2*x^2)/(1-x-x^2)^3: Gser:=series(G,x=0,45): seq(coeff(Gser,x^n),n=1..40); # _Emeric Deutsch_, Feb 07 2006
%p with(combinat): a[1]:=1: a[2]:=0: for n from 3 to 40 do a[n]:=a[n-1]+a[n-2]+n*fibonacci(n-2)-(n-1)*fibonacci(n-3) od: seq(a[n],n=1..40); # _Emeric Deutsch_, May 20 2006
%p A006491:=(z-1)*(1-2*z+2*z**2)/(z**2+z-1)**3; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t LinearRecurrence[{3, 0, -5, 0, 3, 1}, {1, 0, 4, 5, 15, 28}, 50] (* _G. C. Greubel_, Jan 01 2018 *)
%o (PARI) x='x+O('x^30); Vec(x*(1-x)*(1-2*x+2*x^2)/(1-x-x^2)^3) \\ _G. C. Greubel_, Jan 01 2018
%o (Magma) I:=[1,0,4,5,15,28]; [n le 6 select I[n] else 3*Self(n-1) -5*Self(n-3) +3*Self(n-5)+Self(n-6): n in [1..30]]; // _G. C. Greubel_, Jan 01 2018
%Y Cf. A006490, A119458, A320341.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Feb 07 2006
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