0,3

Binomial transform of expansion of cosh(3x), the aerated version of A001019, 1,0,9,0,81,0,729,... - Paul Barry (pbarry(AT)wit.ie), Apr 05 2003

Alternatively: start with the fraction 1/1, take the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top. The limit of the sequence of fractions used to generate this sequence is sqrt(9). - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005

This sequence also gives the number of ordered pairs of subsets (A, B) of {1, 2, ..., n} such that |A u B| is even. (Here "u" stands for the set-theoretic union.) The special case n = 13 can be found as in CRUX Problem 3257. - Walther Janous (walther.janous(AT)tirol.com), Mar 01 2008

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p16

M. Gardner, Riddles of Sphinx, M.A.A., 1987, p. 145.

Bill Sands, Problem 3257, CRUX MATH. 33 (2007), No.5, p. 298.

a(n) = 2*a(n-1) + 8*a(n-2), a(0)=a(1)=1. a(n) = 4^n/2+(-2)^n/2. G.f. (1-x)/((1+2x)(1-4x)). - Paul Barry (pbarry(AT)wit.ie), Mar 01 2003

a(n) := sum{k=0..floor(n/2), C(n, 2k)9^k} E.g.f. exp(x)cosh(3x) - Paul Barry (pbarry(AT)wit.ie), Apr 05 2003

a(n)=(A078008(n)+A001045(n+1))2^(n-1)=A014551(n)*2^(n-1) - Paul Barry (pbarry(AT)wit.ie), Feb 12 2004

Given a(0)=1, b(0)=1 then for i=1, 2, .. a(i)/b(i) =(a(i-1)+ 9*b(i-1)) / (a(i-1) + b(i-1)). - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005

a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*9^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007

((1+sqrt9)^n+(1-sqrt9)^n)/2 in Fibonacci form. [From Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008]

(PARI) g(n, k, typ) = \yp = 1 numerator, 2 denominator, k = multiple of denom { local(a, b, x, tmp); a=1; b=1; for(x=1, n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1, print1(a", "), print1(b", ")) ); print(); print(a/b+.) } - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005

Sequence in context: A126364 A076712 A116973 this_sequence A066527 A103423 A102542

Adjacent sequences: A003662 A003663 A003664 this_sequence A003666 A003667 A003668

nonn

N. J. A. Sloane (njas(AT)research.att.com).

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Nov 22 2006

Corrected A-number in the first comment and added "aerated". - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 23 2008