0,2

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. 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.

Index entries for sequences related to Chebyshev polynomials.

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

(PARI) s(n)=if(n<2, n+1, (s(n-1)+(s(n-2)*2))*2); for(n=0, 32, print(s(n)))

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

nonn

Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001

Better description from Jason Earls (jcearls(AT)cableone.net) and Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 16 2001