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A063727 a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2. 48
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)

Index entries for linear recurrences with constant coefficients, signature (2,4).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

Equals 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/sqrt(4))(-i*sqrt(4))^n, i^2=-1. - Paul Barry, Nov 17 2003

Simplified formula: ((1+sqrt5)^n-(1-sqrt5)^n)/sqrt20. Offset 1. a(3)=8. - Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009

a(n) = 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

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)))

(Sage) [lucas_number1(n, 2, -4) for n in xrange(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

CROSSREFS

Cf. A006483, A103435, A006131.

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 A034741 * A085449 A127362 A133443 A094038

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

STATUS

approved

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Last modified October 17 12:50 EDT 2018. Contains 316280 sequences. (Running on oeis4.)