Search: id:A083098
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%I A083098
%S A083098 1,1,8,22,92,316,1184,4264,15632,56848,207488,756064,2757056,10050496,
%T A083098 36643328,133589632,487039232,1775616256,6473467904,23600633344,
%U A083098 86042074112,313687948288,1143628341248,4169384372224,15200538791936
%N A083098 a(n)=2a(n-1)+6a(n-2).
%C A083098 a(n+1)=a(n)+7*A083099(n-1), a(n+1)/A083099(n) converges to sqrt(7).
%C A083098 Binomial transform of expansion of cosh(sqrt(7)x) (A000420 with interpolated 
               zeros : 1, 0, 7, 0, 49, 0, 343, 0, ...).
%C A083098 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 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 A083098 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%F A083098 G.f.: (1-x)/(1-2x-6x^2).
%F A083098 a(n)=(1+sqrt(7))^n/2+(1-sqrt(7))^n/2 with e.g.f. exp(x)cosh(sqrt(7)x).
%F A083098 a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*7^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 26 2007
%t A083098 CoefficientList[Series[(1+6x)/(1-2x-6x^2), {x, 0, 25}], x]
%o A083098 (Other) sage: [lucas_number2(n,2,-6)/2 for n in xrange(0, 25)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A083098 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 A083098 Sequence in context: A058404 A126362 A140418 this_sequence A033456 A026593 
               A131622
%Y A083098 Adjacent sequences: A083095 A083096 A083097 this_sequence A083099 A083100 
               A083101
%K A083098 easy,nonn
%O A083098 0,3
%A A083098 Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

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