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%I #84 Sep 29 2023 05:29:53
%S 0,2,4,16,48,160,512,1664,5376,17408,56320,182272,589824,1908736,
%T 6176768,19988480,64684032,209321984,677380096,2192048128,7093616640,
%U 22955425792,74285318144,240392339456,777925951488,2517421260800
%N a(n) = 2^n * Fibonacci(n).
%C Cardinality of set of bracelets of size at most n that are tiled with two types of colored squares and four types of colored dominoes.
%C a(n) is also the diagonal element of the matrix A(i,j) whose first row (i=1) and first column (j=1) are the Fibonacci numbers: A(1,k)=A(k,1)=fib(k) and whose generic element is the sum of element in adjacent (preceding) row and column minus the absolute value of their difference. So a(n) = A(n,n) = A(i-1,j)+A(i,j-1)-abs(A(i-1,j)-A(i,j-1)). - _Carmine Suriano_, May 13 2010
%C a(n) is the coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) given for d=sqrt(x+1) by p(n,x)=((x+d)^n-(x-d)^n)/(2d), for n>=1. The constant terms under this reduction are the absolute values of terms of A086344. See A192232 for a discussion of reduction. - _Clark Kimberling_, Jun 29 2011
%C The exponential convolution of A000032 and A000045. - _Vladimir Reshetnikov_, Oct 06 2016
%D Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, identity 236, p. 131.
%H Tom Edgar, <a href="https://www.fq.math.ca/Papers1/54-1/Edgar10292015.pdf">Extending Some Fibonacci-Lucas Relations</a>, The Fibonacci Quarterly, Vol. 54, No. 1 (2016), p. 79.
%H Harris Kwong, <a href="http://dx.doi.org/10.4169/amer.math.monthly.121.06.514">An Alternate Proof of Sury's Fibonacci-Lucas Relation</a>, The American Mathematical Monthly, Vol. 121, No. 6 (2014), p. 514.
%H Diego Marques, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.7.683">A new Fibonacci-Lucas relation</a>, Amer. Math. Monthly, Vol. 122, No. 7 (2015), p. 683.
%H Ivica Martinjak and Helmut Prodinger, <a href="http://arxiv.org/abs/1508.04949">Complementary Families of the Fibonacci-Lucas Relations</a>, arXiv:1508.04949 [math.CO], 2015-2016.
%H B. Sury, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.03.236">A polynomial parent to a Fibonacci-Lucas relations</a>, Amer. Math. Monthly, Vol. 121, No. 3 (2014), p. 236.
%H Charles R. Wall, <a href="https://fq.math.ca/Scanned/25-4/elementary25-4.pdf">Problem B-607</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), p. 370; <a href="https://www.fq.math.ca/Scanned/26-4/elementary26-4.pdf">Product of Exponential Generating Functions</a>, Solution to Problem B-607 by Bob Prielipp, ibid., Vol. 26, No. 4 (1988), pp. 374-375.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,4).
%F a(n) = A006483(n) + 1 = 2*A085449(n) = 2*A063727(n-1), n>0.
%F G.f.: 2*x / (1 - 2*x - 4*x^2).
%F a(n) = Sum_{i=0..n-1}( 2^i * Lucas(i) ).
%F a(n) = 2*a(n-1) + 4*a(n-2). - _Carmine Suriano_, May 13 2010
%F a(n) = a(-n) * -(-4)^n for all n in Z. - _Michael Somos_, Sep 20 2014
%F E.g.f.: 2*sinh(sqrt(5)*x)*exp(x)/sqrt(5). - _Ilya Gutkovskiy_, May 10 2016
%F Sum_{n>=1} 1/a(n) = (1/2) * A269991. - _Amiram Eldar_, Nov 17 2020
%F a(n) == 2*n (mod 10). - _Amiram Eldar_, Jan 15 2022
%F a(n) = Sum_{k=0..n} binomial(n,k) * Fibonacci(k) * Lucas(n-k) (Wall, 1987). - _Amiram Eldar_, Jan 27 2022
%e a(5)=160=A(5,5)=A(4,5)+A(5,4)-abs[A(4,5)+A(5,4)]=80+80-0. - _Carmine Suriano_, May 13 2010
%e G.f. = 2*x + 4*x^2 + 16*x^3 + 48*x^4 + 160*x^5 + 512*x^6 + 1664*x^7 + ...
%t Expand[Table[((1 + Sqrt[5])^n - (1 - Sqrt[5])^n)5/(5 Sqrt[5]), {n, 0, 25}]] (* _Zerinvary Lajos_, Mar 22 2007 *)
%t Table[2^n Fibonacci[n],{n,0,40}] (* or *) LinearRecurrence[{2,4},{0,2},40] (* _Harvey P. Dale_, Oct 14 2020 *)
%o (Magma) [2^n *Fibonacci(n): n in [0..50]]; // _Vincenzo Librandi_, Apr 04 2011
%o (PARI) a(n)=fibonacci(n)<<n \\ _Charles R Greathouse IV_, Feb 03 2014
%o (PARI) concat(0, Vec(2*x/(1-2*x-4*x^2) + O(x^99))) \\ _Altug Alkan_, May 11 2016
%Y First differences of A014334.
%Y Partial sums of A087131.
%Y Cf. A000032, A000045, A006483, A063727, A085449, A269991.
%K nonn,easy
%O 0,2
%A _Ralf Stephan_, Feb 08 2005
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