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 A063727 a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2. 50
 1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160, 91136, 294912, 954368, 3088384, 9994240, 32342016, 104660992, 338690048, 1096024064, 3546808320, 11477712896, 37142659072, 120196169728, 388962975744, 1258710630400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Essentially the same as A085449. Convergents to 2*golden ratio = (1+sqrt(5)). Number of ways to tile an n-board with two types of colored squares and four types of colored dominoes. The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 5 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(5). - Cino Hilliard, Sep 25 2005 a(n) is also the quasi-diagonal element A(i-1,i)=A(1,i-1) of matrix A(i,j) whose elements in first row A(1,k) and first column A(k,1) equal k-th Fibonacci Fib(k) and the generic element is the sum of adjacent (previous) in row and column minus the absolute value of their difference. - Carmine Suriano, May 13 2010 Equals INVERT transform of A006131: (1, 1, 5, 9, 29, 65, 181, ...). - Gary W. Adamson, Aug 12 2010 For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 2's along the three central diagonals. - John M. Campbell, Jul 19 2011 The numbers composing the denominators of the fractional limit to A134972. - Seiichi Kirikami, Mar 06 2012 Pisano period lengths: 1, 1, 8, 1, 5, 8, 48, 1, 24, 5, 10, 8, 42, 48, 40, 1, 72, 24, 18, 5, ... - R. J. Mathar, Aug 10 2012 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 235. John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Harry J. Smith) J. Borowska and L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=2, b=2. Index entries for linear recurrences with constant coefficients, signature (2,4). Index entries for sequences related to Chebyshev polynomials. FORMULA a(n) = 2 * A087206(n+1). From Vladeta Jovovic, Aug 16 2001: (Start) a(n) = sqrt(5)/10*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1)). G.f.: 1/(1-2*x-4*x^2). (End) From Mario Catalani (mario.catalani(AT)unito.it), Jun 13 2003: (Start) a(2*n) = 4*a(n-1)^2 + a(n)^2. A084057(n+1)/a(n) converges to sqrt(5). (End) E.g.f.: exp(x)*(cosh(sqrt(5)*x)+sinh(sqrt(5)*x)/sqrt(5)). - Paul Barry, Sep 20 2003 a(n) = 2^n*Fibonacci(n+1). - Vladeta Jovovic, Oct 25 2003 a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*5^k. - Paul Barry, Nov 15 2003 a(n) = U(n, i/2)*(-i*2)^n, i^2=-1. - Paul Barry, Nov 17 2003 Simplified formula: ((1+sqrt(5))^n-(1-sqrt(5))^n)/sqrt(20). Offset 1. a(3)=8. - Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009 First binomial transform of 1,1,5,5,25,25. - Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009 a(n) = A(n-1,n) = A(n,n-1); A(i,j) = A(i-1,j) + A(i,j-1) - abs(A(i-1,j) - A(i,j-1)). - Carmine Suriano, May 13 2010 G.f.: G(0) where G(k) = 1 + 2*x*(1+2*x)/(1 - 2*x*(1+2*x)/(2*x*(1+2*x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 31 2013 G.f.: G(0)/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013 G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+2 + 4*x )/( x*(4*k+4 + 4*x ) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013 Sum_{n>=0} 1/a(n) = A269991. - Amiram Eldar, Feb 01 2021 MAPLE a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]od: seq(a[n], n=1..33); # Zerinvary Lajos, Dec 15 2008 MATHEMATICA a[n_]:=(MatrixPower[{{1, 5}, {1, 1}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *) CoefficientList[Series[1/(1 - 2 x - 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 31 2014 *) LinearRecurrence[{2, 4}, {1, 2}, 50] (* G. C. Greubel, Jan 07 2018 *) PROG (PARI) s(n)=if(n<2, n+1, (s(n-1)+(s(n-2)*2))*2); for(n=0, 32, print(s(n))) (SageMath) [lucas_number1(n, 2, -4) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009 (PARI) { for (n=0, 200, if (n>1, a=2*a1 + 4*a2; a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); write("b063727.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 28 2009 (Magma) [n le 2 select n else 2*Self(n-1) + 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 07 2018 (GAP) List([0..25], n->2^n*Fibonacci(n+1)); # Muniru A Asiru, Nov 24 2018 CROSSREFS Cf. A006483, A103435, A006131, A269991. Second row of A234357. Row sums of triangle A016095. The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519. Sequence in context: A006952 A327550 A034741 * A085449 A127362 A133443 Adjacent sequences: A063724 A063725 A063726 * A063728 A063729 A063730 KEYWORD nonn,easy AUTHOR Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001 EXTENSIONS Better description from Jason Earls and Vladeta Jovovic, Aug 16 2001 Incorrect comment removed by Greg Dresden, Jun 02 2020 STATUS approved

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Last modified September 17 10:45 EDT 2024. Contains 375987 sequences. (Running on oeis4.)