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 A103435 a(n) = 2^n * Fibonacci(n). 24
 0, 2, 4, 16, 48, 160, 512, 1664, 5376, 17408, 56320, 182272, 589824, 1908736, 6176768, 19988480, 64684032, 209321984, 677380096, 2192048128, 7093616640, 22955425792, 74285318144, 240392339456, 777925951488, 2517421260800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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 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 The exponential convolution of A000032 and A000045. - Vladimir Reshetnikov, Oct 06 2016 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 236. LINKS Tom Edgar, Extending Some Fibonacci-Lucas Relations, Fib. Quarterly, 54 (2016), 79. D. Marques, A new Fibonacci-Lucas relation, Amer. Math. Monthly, 122 (2015), 683. Ivica Martinjak, Complementary Families of the Fibonacci-Lucas Relations, arXiv:1508.04949 [math.CO], 2015. B. Sury, A polynomial parent to a Fibonacci-Lucas relations, Amer. Math. Monthly, 121 (2014), 236. Index entries for linear recurrences with constant coefficients, signature (2,4). FORMULA G.f.: 2*x / (1 - 2*x - 4*x^2). a(n) = Sum_{i=0..n-1}( 2^i * Lucas(i) ). a(n) = 2*a(n-1) + 4*a(n-2). - Carmine Suriano, May 13 2010 a(n) = a(-n) * -(-4)^n for all n in Z. - Michael Somos, Sep 20 2014 E.g.f.: 2*sinh(sqrt(5)*x)*exp(x)/sqrt(5). - Ilya Gutkovskiy, May 10 2016 EXAMPLE 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 G.f. = 2*x + 4*x^2 + 16*x^3 + 48*x^4 + 160*x^5 + 512*x^6 + 1664*x^7 + ... MATHEMATICA Expand[Table[((1 + Sqrt)^n - (1 - Sqrt)^n)5/(5 Sqrt), {n, 0, 25}]] (* Zerinvary Lajos, Mar 22 2007 *) PROG (MAGMA) [2^n *Fibonacci(n): n in [0..50]]; // Vincenzo Librandi, Apr 04 2011 (PARI) a(n)=fibonacci(n)<0. First differences of A014334. Partial sums of A087131. Sequence in context: A112638 A077162 A128903 * A119000 A034917 A215724 Adjacent sequences:  A103432 A103433 A103434 * A103436 A103437 A103438 KEYWORD nonn,easy AUTHOR Ralf Stephan, Feb 08 2005 STATUS approved

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)