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A063727
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a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
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34
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Essentially the same as A085449.
Convergents to golden ratio (1+sqrt(5))/2.
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 (hillcino368(AT)gmail.com), 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 adiacent (previous) in row and column minus the absolute value of their difference [From Carmine Suriano, May 13 2010]
Equals INVERT transform of A006131: (1, 1, 5, 9, 29, 65, 181,...). [From Gary W. Adamson, Aug 12 2010]
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 2's along the three central diagonals. [From John M. Campbell, Jul 19 2011]
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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.
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LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,200
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,4).
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n) = sqrt(5)/10*((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1)). G.f.: 1/(1-2*x-4*x^2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 16 2001
a(2*n)=4*a(n-1)^2+a(n)^2. A084057(n+1)/a(n) converges to sqrt(5). - Mario Catalani (mario.catalani(AT)unito.it), Jun 13 2003
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 (vladeta(AT)eunet.rs), 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 [From Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009]
a(n)=first binomial trnasform of 1,1,5,5,25,25 [From 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)] [From Carmine Suriano, May 13 2010]
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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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 15 2008]
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MATHEMATICA
| a[n_]:=(MatrixPower[{{1, 5}, {1, 1}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] [From Vladimir Orlovsky, Feb 19 2010]
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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)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Aug 28 2009]
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CROSSREFS
| Equals 2 * A087206(n+1). Cf. A006483.
Row sums of triangle A016095.
Cf. A103435.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A103435 [From Carmine Suriano, May 13 2010]
Cf. A006131 [From Gary W. Adamson, Aug 12 2010]
Sequence in context: A006952 A034741 * A085449 A127362 A133443 A094038
Adjacent sequences: A063724 A063725 A063726 * A063728 A063729 A063730
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KEYWORD
| nonn,easy
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AUTHOR
| Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001
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EXTENSIONS
| Better description from Jason Earls (zevi_35711(AT)yahoo.com) and Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 16 2001
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