Search: id:A079935
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%I A079935
%S A079935 1,3,11,41,153,571,2131,7953,29681,110771,413403,1542841,5757961,
%T A079935 21489003,80198051,299303201,1117014753,4168755811,15558008491,
%U A079935 58063278153,216695104121,808717138331,3018173449203,11263976658481
%N A079935 a(n) = 4a(n-1) - a(n-2).
%C A079935 See A001835 for another version.
%C A079935 Greedy frac multiples of sqrt(3): a(1)=1, sum(n>0,frac(a(n)*x)) < 1 at 
               x=sqrt(3).
%C A079935 The n-th greedy frac multiple of x is the smallest integer that does 
               not cause sum(k=1..n,frac(a(k)*x)) to exceed unity; an infinite number 
               of terms appear as the denominators of the convergents to the continued 
               fraction of x.
%C A079935 In general, sum{k=0..n, binomial(2n-k,k)j^(n-k)}=(-1)^n*U(2n,I*sqrt(j)/
               2), I=sqrt(-1). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005
%C A079935 The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,0,0,...]. 
               - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007
%D A079935 Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica 
               Journal, 9:3 (2005), 609-640.
%H A079935 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A079935 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%F A079935 For n>0, a(n)= ceil( (2+sqrt(3))^n/(3+sqrt(3)) ).
%F A079935 G.f.: (1-x)/(1-4x+x^2); E.g.f.: exp(2x)(sinh(sqrt(3)x)/sqrt(3)+cosh(sqrt(3)x)); 
               a(n)=(1/2+sqrt(3)/6)(2+sqrt(3))^n+(1/2-sqrt(3)/6)(2-sqrt(3))^n (offset 
               0). Binomial transform of A002605. - Paul Barry (pbarry(AT)wit.ie), 
               Sep 17 2003
%F A079935 a(n)=sum{k=0..n, binomial(2n-k, k)2^(n-k)}; - Paul Barry (pbarry(AT)wit.ie), 
               Jan 22 2005
%F A079935 a(n)=(-1)^n*U(2n, I*sqrt(2)/2), U(n, x) Chebyshev polynomial of second 
               kind, I=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005
%F A079935 a(n)=Jacobi_P(n,-1/2,1/2,2)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), 
               Feb 03 2006
%e A079935 a(4) = 41 since frac(1x) + frac(3x) + frac(11x) + frac(41x) < 1, while 
               frac(1x) + frac(3x) + frac(11x) + frac(k*x) > 1 for all k>11 and 
               k<41.
%t A079935 a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[1, 1]]; Table[ 
               a[n], {n, 0, 23}]] (from Robert G. Wilson v Jan 13 2005)
%o A079935 (Other) sage: [lucas_number1(n,4,1)-lucas_number1(n-1,4,1) for n in xrange(1, 
               25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 
               2009]
%Y A079935 Cf. A002530 (denominators of convergents to sqrt(3)), A079934, A079936, 
               A001353.
%Y A079935 Cf. A001835 (same except for the first term).
%Y A079935 Row 4 of array A094954.
%Y A079935 Sequence in context: A077831 A032952 A001835 this_sequence A113437 A076540 
               A129637
%Y A079935 Adjacent sequences: A079932 A079933 A079934 this_sequence A079936 A079937 
               A079938
%K A079935 nonn
%O A079935 1,2
%A A079935 Benoit Cloitre (benoit7848c(AT)orange.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), 
               Jan 20 2003

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