0,3

a(n)=a(n-1)+A083100(n-2), n>1. A083100(n)/a(n+1) converges to sqrt(8). - Mario Catalani (mario.catalani(AT)unito.it), Apr 23 2003

The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 8 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(8). - Cino Hilliard (hillcino368(AT)gmail.com), Sep 25 2005

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

G.f.: x/((1-(1+sqrt(8))x)(1-(1-sqrt(8))x)); a(n) := ((1+2sqrt(2))^n-(1-2sqrt(2))^n)sqrt(2)/8. - Paul Barry (pbarry(AT)wit.ie), Jul 17 2003

E.g.f. : exp(x)sinh(2sqrt(2)x)/(2sqrt(2)). - Paul Barry (pbarry(AT)wit.ie), Nov 20 2003

Second binomial transform is A000129(2n)/2 (A001109). - Paul Barry (pbarry(AT)wit.ie), Apr 21 2004

a(n)=sum(comb(n-k-1, k)(7/2)^k2^(n-k-1), k, 0, floor((n-1)/2)). - Paul Barry (pbarry(AT)wit.ie), Jul 17 2004

a(n)=sum{k=0..n, binomial(n, 2k+1)8^k} - Paul Barry (pbarry(AT)wit.ie), Sep 29 2004

(Other) sage: [lucas_number1(n, 2, -7) for n in xrange(0, 25)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

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: A015516 A015517 A015518 this_sequence A015520 A015521 A015522

Sequence in context: A154416 A071244 A005583 this_sequence A096977 A084098 A152819

nonn,easy

Olivier Gerard (olivier.gerard(AT)gmail.com)