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%I #104 Jan 31 2023 05:58:22
%S 1,2,8,24,80,256,832,2688,8704,28160,91136,294912,954368,3088384,
%T 9994240,32342016,104660992,338690048,1096024064,3546808320,
%U 11477712896,37142659072,120196169728,388962975744,1258710630400
%N a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
%C Essentially the same as A085449.
%C Convergents to 2*golden ratio = (1+sqrt(5)).
%C Number of ways to tile an n-board with two types of colored squares and four types of colored dominoes.
%C 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
%C 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
%C Equals INVERT transform of A006131: (1, 1, 5, 9, 29, 65, 181, ...). - _Gary W. Adamson_, Aug 12 2010
%C 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
%C The numbers composing the denominators of the fractional limit to A134972. - _Seiichi Kirikami_, Mar 06 2012
%C 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
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 235.
%D John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
%H G. C. Greubel, <a href="/A063727/b063727.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Harry J. Smith)
%H J. Borowska, L. Lacinska, <a href="https://doi.org/10.17512/jamcm.2014.4.03">Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix</a>, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=2, b=2.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,4).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = 2 * A087206(n+1).
%F From _Vladeta Jovovic_, Aug 16 2001: (Start)
%F a(n) = sqrt(5)/10*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1)).
%F G.f.: 1/(1-2*x-4*x^2). (End)
%F From Mario Catalani (mario.catalani(AT)unito.it), Jun 13 2003: (Start)
%F a(2*n) = 4*a(n-1)^2 + a(n)^2.
%F A084057(n+1)/a(n) converges to sqrt(5). (End)
%F E.g.f.: exp(x)*(cosh(sqrt(5)*x))+sinh(sqrt(5)*x)/sqrt(5)). - _Paul Barry_, Sep 20 2003
%F a(n) = 2^n*Fibonacci(n+1). - _Vladeta Jovovic_, Oct 25 2003
%F a(n) = sum{k=0..floor(n/2), C(n, 2*k+1)*5^k}. - _Paul Barry_, Nov 15 2003
%F a(n) = U(n, i/sqrt(4))(-i*sqrt(4))^n, i^2=-1. - _Paul Barry_, Nov 17 2003
%F Simplified formula: ((1+sqrt5)^n-(1-sqrt5)^n)/sqrt20. Offset 1. a(3)=8. - Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009
%F First binomial transform of 1,1,5,5,25,25. - Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009
%F 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
%F 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
%F 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
%F 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
%F Sum_{n>=0} 1/a(n) = A269991. - _Amiram Eldar_, Feb 01 2021
%p 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
%t 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 *)
%t CoefficientList[Series[1/(1 - 2 x - 4 x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 31 2014 *)
%t LinearRecurrence[{2, 4}, {1, 2}, 50] (* _G. C. Greubel_, Jan 07 2018 *)
%o (PARI) s(n)=if(n<2,n+1,(s(n-1)+(s(n-2)*2))*2); for(n=0,32,print(s(n)))
%o (Sage) [lucas_number1(n,2,-4) for n in range(1, 26)] # _Zerinvary Lajos_, Apr 22 2009
%o (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
%o (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
%o (GAP) List([0..25],n->2^n*Fibonacci(n+1)); # _Muniru A Asiru_, Nov 24 2018
%Y Cf. A006483, A103435, A006131, A269991.
%Y Second row of A234357. Row sums of triangle A016095.
%Y The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
%K nonn,easy
%O 0,2
%A Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001
%E Better description from _Jason Earls_ and _Vladeta Jovovic_, Aug 16 2001
%E Incorrect comment removed by _Greg Dresden_, Jun 02 2020
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