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Search: id:A063727
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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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+0 27
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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; listen)
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OFFSET
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0,2
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COMMENT
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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
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REFERENCES
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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
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Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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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.yu), Aug 16 2001
a(2n)=4a(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 (pbarry(AT)wit.ie), Sep 20 2003
a(n) = 2^n*Fibonacci(n+1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 25 2003
a(n)=sum{k=0..floor(n/2), C(n, 2k+1)5^k} - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003
a(n)=U(n, i/sqrt(4))(-i*sqrt(4))^n, i^2=-1 - Paul Barry (pbarry(AT)wit.ie), Nov 17 2003
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PROGRAM
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(PARI) s(n)=if(n<2, n+1, (s(n-1)+(s(n-2)*2))*2); for(n=0, 32, print(s(n)))
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CROSSREFS
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Essentially the same as A085449.
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.
Adjacent sequences: A063724 A063725 A063726 this_sequence A063728 A063729 A063730
Sequence in context: A006952 A034741 A085449 this_sequence A127362 A133443 A094038
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KEYWORD
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nonn
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AUTHOR
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Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001
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EXTENSIONS
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Better description from Jason Earls (jcearls(AT)cableone.net) and Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 16 2001
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