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.

Harry J. Smith, Table of n, a(n) for n=0,...,200

Index entries for sequences related to linear recurrences with constant coefficients

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.rs), 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.rs), 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

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[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]

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

sage: from sage.combinat.sloane_functions import recur_gen2b sage: it = recur_gen2b(1, 2, 2, 4, lambda n: 0) sage: [it.next() for i in xrange(1, 26)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 03 2008

(Other) 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]

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 (zevi_35711(AT)yahoo.com) and Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 16 2001