%I A083099
%S A083099 0,1,2,10,32,124,440,1624,5888,21520,78368,285856,1041920,3798976,
%T A083099 13849472,50492800,184082432,671121664,2446737920,8920205824,
%U A083099 32520839168,118562913280,432250861568,1575879202816,5745263575040
%N A083099 a(0) = 0, a(1) = 1; for n>1, a(n) = 2a(n-1)+6a(n-2).
%C A083099 a(n+1) = a(n)+A083098(n+1). A083098(n+1)/a(n) converges to sqrt(7).
%C A083099 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 7 times the bottom to get the new top. The limit of the 
               sequence of fractions is sqrt(7). - Cino Hilliard (hillcino368(AT)gmail.com), 
               Sep 25 2005
%D A083099 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%F A083099 G.f.: 1/(1-2x-6x^2).
%F A083099 E.g.f. : dif(exp(x)sinh(sqrt(7)x)/sqrt(7), x); a(n-1)=sum{k=0..n, binomial(n, 
               2k+1)7^k}. - Paul Barry (pbarry(AT)wit.ie), Sep 29 2004
%F A083099 a(n)=-(1/14)*[1-sqrt(7)]^n*sqrt(7)+(1/14)*[1+sqrt(7)]^n*sqrt(7), with 
               n>=0 - Paolo P. Lava (ppl(AT)spl.at), Jun 10 2008
%F A083099 Simplified formula: ((1+sqrt7)^n-(1-sqrt7)^n)/sqrt28. Offset 1. a(3)=10 
               [From Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009]
%t A083099 CoefficientList[Series[1/(1-2x-6x^2), {x, 0, 25}], x]
%t A083099 Expand[Table[((1 + Sqrt[7])^n - (1 - Sqrt[7])^n)7/(14Sqrt[7]), {n, 0, 
               25}]] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 22 2007
%o A083099 (Other) sage: [lucas_number1(n,2,-6) for n in xrange(0, 25)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%Y A083099 The following sequences (and others) belong to the same family: A001333, 
               A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, 
               A002532, A083098, A083099, A083100, A015519.
%Y A083099 Sequence in context: A131068 A034555 A084154 this_sequence A032095 A151019 
               A004028
%Y A083099 Adjacent sequences: A083096 A083097 A083098 this_sequence A083100 A083101 
               A083102
%K A083099 easy,nonn
%O A083099 0,3
%A A083099 Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003